Log-Concavity of Recursively Defined Polynomials

Bernhard Heim and Markus Neuhauser Department of Mathematics and Science Faculty of Science German University of Technology in Oman (GUtech) PO Box 1816 Athaibah PC 130 Sultanate of Oman

              bernhard.heim@gutech.edu.om
            markus.neuhauser@gutech.edu.om

Abstract

Fourier coefficients of powers of the Dedekind eta function can be studied by polynomials introduced by M. Newman. We generalize the defining recurrence relations in this paper. From this we derive new families of polynomials, which approximate these polynomials from below and above. Although these families are recursively defined, we are able to determine explicit closed formulas for both approximating polynomials. (For the original polynomials closed formulas are not yet known.) Furthermore, we obtain that both approximating families and the coefficients involved are log-concave and unimodal.

1 Introduction

Let η(τ) be the Dedekind eta function [7]. Put q := e^{2π i τ}, where τ is in the upper half-space ℍ. Let further z ∈ ℂ. We consider the Fourier expansion

(q^{− \frac{1}{24}} η(τ))^{−z} = ∏_{n=1}^{∞} (1 − q^{n})^{−z} = ∑_{n=0}^{∞} P_{n}(z) q^{n}.  (1)

We recover several famous sequences [2, Introduction]. For example, the partition numbers p(n) and the Ramanujan numbers τ(n) are directly linked to z = 1 and z = −24,

(P_{n}(1))_{n=0}^{∞} = (p(n))_{n=0}^{∞} : 1, 1, 2, 3, 5, … (\underline{A000041});
(P_{n−1}(−24))_{n=1}^{∞} = (τ(n))_{n=1}^{∞} : 1, −24, 252, … (\underline{A000594}).

The integer-valued polynomials P_{n}(X) have degree n and positive integral coefficients A_{k}^{n} for 0 ≤ k ≤ n − 1:

P_{n}(X) = \frac{X}{n!} ∑_{k=0}^{n−1} A_{k}^{n} X^{k}.  (2)

Newman [5] calculated the coefficients recursively for 0 ≤ k ≤ n − 1. For example A_{0}^{n} = (n − 1)!σ(n), A_{n−1}^{n} = 1, and A_{n−2}^{n} = 3n(n − 1)/2. Here σ(n) is the sum of the divisors of n. Further Newman determined the first ten polynomials and their integral zeros and found a recursion formula for the involved polynomials. Apparently there is yet no closed explicit formula for these polynomials defined by

P_{n}(X) = \frac{X}{n} ∑_{k=1}^{n} σ(k)P_{n−k}(X)  (3)

and the involved coefficients.

Numerical calculations indicate that the sequence of the coefficients A_{k}^{n} seems to be close to unimodal and stable (Hurwitz polynomial). Here we normalize P_{n}(X) by n!/X. A direct approach seems to be out of reach at the moment, since there is still a lack of understanding the properties of the arithmetic function σ(n) completely. Nevertheless the value distribution of the polynomials P_{n}(X), especially the zeros, have significant applications. For r even and positive, the sequences (P_{n}(−r))_{n} are lacunary if and only if r = 2, 4, 6, 8, 10, 14, 26. This result is due to Serre [8] and is in relation to modular forms with complex multiplication (CM-forms). The non-existence of −24 as a zero is equivalent to the Lehmer conjecture [4].

In this paper we generalize the recurrence relation (3), studying closed formulas for the polynomials associated with the functions g(n) = n and g(n) = n^{2}. They are integer valued and approximate σ(n) from below and above.

Definition 1. Let g(n) be an arithmetic function. We define a family of polynomials P_{n}^{g}(X) associated with g. Let P_{0}^{g}(X) := 1 and

P_{n}^{g}(X) := \frac{X}{n} ∑_{k=1}^{n} g(k) P_{n−k}^{g}(X),  (4)

= \frac{X}{n!} ∑_{k=0}^{n−1} A_{k}^{n}(g) X^{k}.  (5)

Let φ_{1}(n) = n and φ_{2}(n) = n^{2}. In what follows, we study the properties of the associated polynomials P_{n}^{φ_{1}}(X) and P_{n}^{φ_{2}}(X). Their properties are related to P_{n}(X) = P_{n}^{σ}(X), since φ_{1}(n) < σ(n) < φ_{2}(n) for n > 1. We obtain A_{n−1}^{n}(φ_{1}) = A_{n−1}^{n} = A_{n−1}^{n} (φ_{2}) = 1. Further, let 0 ≤ k ≤ n − 2. Then

A_{k}^{n}(φ_{1}) < A_{k}^{n} < A_{k}^{n}(φ_{2}).  (6)

Theorem 2. Let φ_{1}(n) = n. Then the coefficients of P_{n}^{φ_{1}}(X) are given by

A_{k}^{n}(φ_{1}) = \frac{n!}{(k + 1)!} \binom{n − 1}{k}.  (7)

Although the binomial coefficients are twisted by 1/(k + 1)!, they remain log-concave.

Corollary 3. The sequence of the coefficients of P_{n}^{φ_{1}}(X)

(A_{k}^{n}(φ_{1}))_{k=0}^{n−1}

is strongly log-concave and hence unimodal.

Further we can determine the index, and therefore the size of the largest coefficient.

Corollary 4. The index K_{1} of the maximal coefficient A_{k}^{n}(φ_{1}) is given by

\sqrt{n + 1} − 2 ≤ K_{1} < \sqrt{n + 1} − 1.  (8)

In general, K_{1} is not always unique.

Numerical calculations indicate that the zeros of P_{n}^{φ_{1}}(X) are simple and that the polynomials are stable in the sense of Hurwitz (neglecting X = 0).

It is also possible to get similar results for P_{n}^{φ_{2}}(X), although the involved coefficients A_{k}^{n}(φ_{2}) are more complicated. We also studied the polynomials P_{n}^{φ}(X) associated with φ(n) := n ln(n) and φ(n) := n \sqrt{n}. But a closed formula for the coefficients is very difficult to obtain.

Theorem 5. Let φ_{2}(n) = n^{2}. Then the coefficients of P_{n}^{φ_{2}}(X) are given by

A_{k}^{n}(φ_{2}) = \frac{n!}{(k + 1)!} \binom{n + k}{2k + 1} .  (9)

Corollary 6. The sequence of the coefficients of P_{n}^{φ_{2}}(X)

(A_{k}^{n}(φ_{2}))_{k=0}^{n−1}

is strongly log-concave and hence unimodal.

Corollary 7. The coefficients A_{k}^{n}(φ_{2}) assume their maximum at the index K_{2}, such that K_{2} ≤ k < K_{2} + 1 is the real solution of the cubic equation 4k^{3} + 19k^{2} + 28k + 13 = n^{2}. Let

D := \frac{1}{8}n^{2} − \frac{91}{1728} + \frac{1}{144}\sqrt{324n^{4} − 273n^{2} − 51},

then \sqrt[3]{D} + \frac{25}{144 \sqrt[3]{D}} ≤ K_{2} < \sqrt[3]{D} + \frac{25}{144 \sqrt[3]{D}} + 1. For large n we obtain K_{2} ≈ (n/2)^{2/3}.

Numerical calculations indicate that the zeros of P_{n}^{φ_{1}}(X) are simple and that the polynomials are stable.

Let x_{0} be a positive real number. Let g_{1}, g_{2} be two arithmetic functions, satisfying 1 ≤ g_{1}(n) ≤ g_{2}(n). Then P_{n}^{g_{1}}(x_{0}) ≤ P_{n}^{g_{2}}(x_{0}). We let p(n) denote the partition numbers [1]. As an application we obtain an approximation of the partition numbers from below. Let N = 200. Then

\frac{p(n)}{P_{n}^{φ_{1}}(1)} < 647.71  for n ≤ N.  (10)

Finally we obtain for x = −1 the following result.

Figure 1: ‘+’ marks the value of P_{n}^{φ_{1}} (−1), ‘×’ marks the value of P_{n}^{σ} (−1) depending on n.

Euler showed that P_{n}(−1) takes only the values −1, 0, 1 and has high vanishing rate, called superlacunary [6] in the language of modular forms. This is reflected by the asymptotic behavior of the values of the sequence P_{n}^{φ_{1}}(−1), although x_{0} < 0.

2 Log-Concavity and maximal coefficients

A sequence a_{0}, a_{1}, a_{2}, …, a_{n} of real numbers is called unimodal if the sequence increases steadily at first and then decreases steadily [9]. The sequence of binomial coefficients and Stirling numbers are unimodal. In the case where

a_{0} ≤ a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{K} ≥ a_{K+1} ≥ a_{K+2} ≥ ⋯ ≥ a_{m},  (11)

the K is called the index of the sequence. The index does not have to be unique. In general, it is not clear how to determine the index.

Another important property of some sequences is (strong) log-concavity. A sequence is called log-concave if for all k = 1, 2, …, m − 1:

a_{k}^{2} ≥ a_{k+1} a_{k−1}.  (12)

Note that log-concavity implies unimodality.

Due to Newton’s inequality, the sequence of coefficients of a polynomial with (real coefficients and) only real zeros is strongly log-concave. It is very likely that P_{n}^{φ_{1}}(X) and P_{n}^{φ_{2}}(X) have only real zeros. But P_{n}(X) definitely does not have only real zeros. Heim et al. [3] showed that, for example, in the case n = 10 non-real zeros appear. Nevertheless, directly applying the log-concave criterion works for φ_{1}, φ_{2}, due to the explicit formulas proved in this paper.

2.1 Proof of Corollary 3

Let n ≥ 3 be given. We consider the sequence

a_{k} := A_{k}^{n}(φ_{1}) = \frac{n!}{(k + 1)!} \binom{n − 1}{k}

for k = 0, …, n − 1. We show that for each pair (n, k) with 1 ≤ k ≤ n − 2 an α > 1 exists, such that

\frac{A_{k}^{n}(φ_{1})^{2}}{A_{k+1}^{n}(φ_{1}) A_{k−1}^{n}(φ_{1})} ≥ α.  (13)

This quotient results in

\begin{matrix}(\frac{1}{(k+1)!} \binom{n−1}{k}) & 2 \\ \frac{1}{k!} & \binom{n−1}{k−1} & \frac{1}{(k+2)!} & \binom{n−1}{(k+1)} \end{matrix} = ^{\begin{matrix}\frac{1}{k+1} & (\frac{(n−1)!}{k!(n−k−1)!}) & 2 \\ \frac{(n−1)!}{(k−1)!(n−k)!} & \frac{1}{k+2} & \frac{(n−1)!}{(k+1)!(n−k−2)!} \end{matrix}}

= \begin{matrix}\frac{1}{k+1} & \frac{1}{k(n−k−1)} \\ \frac{1}{n−k} & \frac{1}{k+2} & \frac{1}{k+1} \end{matrix}
= \frac{(n − k) (k + 2)}{k (n − k − 1)} ≥ α > 1.

Hence the sequence A_{k}^{n}(φ_{1}) is strongly log-concave. This implies unimodality.

We obtain α = α_{n} ≥ 1+(3n + 1 + 2\sqrt{2n (n + 1)}) (n − 1)^{−2}. Depending on n α decreases to 1.

To show that α decreases to 1, consider the denominator of the derivative of \frac{(n−k)(k+2)}{k(n−1−k)} with respect to k. The denominator yields 2n (n + 1) − (k − 2n)^{2}. Hence the minimum of the denominator for 1 ≤ k ≤ n − 2 is obtained at k = 2n − \sqrt{2n (n + 1)}. This yields the claimed value.

2.2 Proof of Corollary 6

Let n ≥ 2 be given. We consider the sequence

a_{k} := A_{k}^{n}(φ_{2}) = \frac{n!}{(k + 1)!} \binom{n + k}{2k + 1}

for k = 0, …, n − 1. Hence a_{0}, a_{1}, …, a_{m} with m = n − 1. We show that for each pair (n, k) with 0 ≤ k ≤ n − 2 a β > 1 exists such that

\frac{A_{k}^{n}(φ_{2})^{2}}{A_{k+1}^{n}(φ_{2}) A_{k−1}^{n}(φ_{2})} > β.  (14)

This quotient results in

\begin{matrix}(\frac{1}{(k+1)!} \binom{n+k}{2k+1}) & 2 \\ \frac{1}{k!} & \binom{n+k−1}{2k−1} & \frac{1}{(k+2)!} & \binom{n+k+1}{2k+3} \end{matrix} = ^{\begin{matrix}\frac{1}{k+1} & (\frac{(n+k)!}{(2k+1)!(n−k−1)!}) & 2 \\ \frac{(n+k−1)!}{(2k−1)!(n−k)!} & \frac{1}{k+2} & \frac{(n+k+1)!}{(2k+3)!(n−k−2)!} \end{matrix}}

= \begin{matrix}\frac{1}{k+1} & \frac{n+k}{(2k+1)2k(n−k−1)} \\ \frac{1}{n−k} & \frac{1}{k+2} & \frac{n+k+1}{(2k+3)(2k+2)} \end{matrix}
= \frac{(n + k) (n − k) (k + 2) (2k + 3) (2k + 2)}{(k + 1) (n + k + 1) (2k + 1) 2k (n − k − 1)}
> \frac{n^{2} − k^{2}}{n^{2} − (k + 1)^{2}} ≥ β > 1.

Hence the sequence A_{k}^{n}(φ_{2}) is strongly log-concave. This implies unimodality.

We have \frac{n^{2}−k^{2}}{n^{2}−(k+1)^{2}} = 1 + \frac{2k+1}{n^{2}(k+1)^{2}} . This is increasing as a function of k for 1 ≤ k ≤ n − 2. So the minimum is attained at k = 1 and results in 1 + \frac{3}{n^{2}−4}. For n → ∞ 1 + \frac{3}{n^{2}−4} again decreases to 1. Even if we could determine the minimum of the expression before the ‘strictly less’ sign, we would obtain the same limiting behavior.

2.3 Proof of Corollary 4

The idea is simple. Let n ≥ 2. Let a_{k} = A_{k}^{n}(φ_{1}), 0 ≤ k ≤ n − 1. We analyze the possible sign changes of

Δ_{k}(φ_{1}) := a_{k+1} − a_{k} = A_{k+1}^{n}(φ_{1}) − A_{k}^{n}(φ_{1}), 0 ≤ k ≤ n − 2.  (15)

We obtain

Δ_{k}(φ_{1}) = γ(k, n) (\frac{1}{(k + 2) (k + 1)} − \frac{1}{n − k − 1}) ,  (16)

where γ(k, n) is a positive rational number. Hence Δ_{k}(φ_{1}) has the same sign as

n + 1 − (k + 2)^{2}.

Hence the coefficients increase for k < \sqrt{n + 1} − 2, and decrease for k > \sqrt{n + 1} − 2. In the case of equality we have two maxima.

2.4 Proof of Corollary 7

Let n ≥ 2. Let a_{k} = A_{k}^{n}(φ_{2}), 0 ≤ k ≤ n − 1. We analyze the possible sign changes of

Δ_{k}(φ_{2}) := a_{k+1} − a_{k} = A_{k+1}^{n}(φ_{2}) − A_{k}^{n}(φ_{2}), 0 ≤ k ≤ n − 2.  (17)

We obtain

Δ_{k}(φ_{2}) = γ′(k, n) (\frac{n + k + 1}{(k + 2) (2k + 3) (2k + 2)} − \frac{1}{n − k − 1}) .  (18)

where γ′(k, n) is a positive rational number. Hence Δ_{k}(φ_{2}) has the same sign as

−4k^{3} − 19k^{2} − 28k − 13 + n^{2}.

As a function of k ≥ 0, the expression Δ_{k}(φ_{2}) is decreasing. Hence there exists exactly one 0 ≤ K ≤ n − 1 such that

A_{K−1}^{n}(φ_{2}) < A_{K}^{n}(φ_{2}) ≥ A_{K+1}^{n}(φ_{2}).

We used the computer algebra system Maple to calculate the algebraic expression defining K. Nevertheless, it is obvious that K ≈ 2^{−2/3} n^{\frac{2}{3}} for large n.

3 Explicit formulas for P_{n}^{φ_{1}}(X)

Let us start with a list of the polynomials for n = 1, 2, …, 7. Note that they are no longer integer-valued polynomials, although the modified coefficients are integral.

P_{1}^{φ_{1}}(X) = X
P_{2}^{φ_{1}}(X) = \frac{1}{2}X (X + 2)
P_{3}^{φ_{1}}(X) = \frac{1}{6}X (X^{2} + 6X + 6)
P_{4}^{φ_{1}}(X) = \frac{1}{24}X (X^{3} + 12X^{2} + 36X + 24)
P_{5}^{φ_{1}}(X) = \frac{1}{120}X (X^{4} + 20X^{3} + 120X^{2} + 240X + 120)

P_{6}^{φ_{1}}(X) = \frac{1}{720}X (X^{5} + 30X^{4} + 300X^{3} + 1200X^{2} + 1800X + 720)
P_{7}^{φ_{1}}(X) = \frac{1}{5040}X (X^{6} + 42X^{5} + 630X^{4} + 4200X^{3} + 12600X^{2} + 15120X + 5040) .

Remark 8. The first polynomials \frac{n!}{X}P_{n}^{φ_{1}}(X) are irreducible. Proof of Theorem 2. We start with the identity

∑_{n=0}^{∞} P_{n}^{φ_{1}} (X) q^{n} = exp (X ∑_{n=1}^{∞} q^{n}) .  (19)

It is useful to substitute (∑_{n=1}^{∞} q^{n})^{k} by the multi-index sum

∑_{m_{1},...,m_{k}=1}^{∞} q^{m_{1}+···+m_{k}}.

Comparing the coefficients of q^{n} in (19) leads to

P_{n} (X) = ∑_{k=1}^{n} \frac{1}{k!} ( ∑_{m_{1}+···+m_{k}=n} 1) X^{k}
= \frac{X}{n!} ∑_{k=0}^{n−1} \frac{n!}{(k + 1)!} ( ∑_{m_{1}+···+m_{k+1}=n} 1) X^{k}.

Note that

∑_{m_{1}+···+m_{k}=n} 1 = \binom{n − 1}{k − 1}.  (20)

We prove (20) by induction. For k = 1 we have 1 = \binom{n−1}{0} = \binom{n−1}{k−1}. Suppose the formula holds for k then

∑_{m_{1}+···+m_{k+1}=n} 1 = ∑_{m_{k+1}=1}^{n−k} ∑_{m_{1}+···+m_{k}=n−m_{k+1}} 1 = ∑_{m_{k+1}=1}^{n−k} \binom{n − m_{k+1} − 1}{k − 1}
= ∑_{m_{k+1}=k}^{n−1} \binom{m_{k+1} − 1}{k − 1} = \binom{n − 1}{k}.

The proof (again by induction) of the identity ∑_{m=k}^{n−1} \binom{m−1}{k−1} = \binom{n−1}{k} we leave to the reader.

4 Explicit formulas for P_{n}^{φ_{2}}(X)

Let us start with a list of the polynomials for n = 1, 2, … 7. Note that they are no longer integer-valued polynomials, although the modified coefficients are integral.

P_{1}^{φ_{2}}(X) = X
P_{2}^{φ_{2}}(X) = 1/2 X (4 + X)
P_{3}^{φ_{2}}(X) = 1/6 X (X^{2} + 12 X + 18)
P_{4}^{φ_{2}}(X) = 1/24 X (X^{3} + 24 X^{2} + 120 X + 96)
P_{5}^{φ_{2}}(X) = \frac{1}{120}X (X^{4} + 40 X^{3} + 420 X^{2} + 1200 X + 600)
P_{6}^{φ_{2}}(X) = \frac{1}{720}X (X^{5} + 60 X^{4} + 1080 X^{3} + 6720 X^{2} + 12600 X + 4320)
P_{7}^{φ_{2}}(X) = \frac{1}{5040}X (X^{6} + 84 X^{5} + 2310 X^{4} + 25200 X^{3} + 105840 X^{2} + 141120 X + 35280)

Before we prove Theorem 5 we show the following useful property.

Lemma 9. For n ≥ 1 and 1 ≤ k ≤ n we obtain

∑_{m_{1}+···+m_{k}=n} m_{1}m_{2} ⋯ m_{k} = \binom{n + k − 1}{2k − 1}.  (21)

Proof. The proof is by induction on n and k. For k = 1 we have n = \binom{n}{1} = \binom{n+k−1}{2k−1}. Let now N ≥ K ≥ 2 be fixed. We assume that formula (21) holds for k < K. For k = K we also assume that the formula (21) holds for n < N. Then we will prove formula (21) for n = N and k = K. We obtain for

∑_{m_{1}+···+m_{K}=N} m_{1}m_{2} ⋯ m_{K}

the expressions

∑_{m_{K}=1}^{N+1−K} m_{K} ∑_{m_{1}+···+m_{K−1}=N−m_{K}} m_{1}m_{2} ⋯ m_{K−1}

= ∑_{m_{K}=1}^{N+1−K} ∑_{\begin{matrix}m_{1}+···+m_{K−1} \\ =N−m_{K} \end{matrix}} m_{1} ⋯ m_{K−1} + ∑_{m_{K}=1}^{N−K} m_{K} ∑_{\begin{matrix}m_{1}+···+m_{K−1} \\ =N−1−m_{K} \end{matrix}} m_{1}m_{2} ⋯ m_{K−1}

= ∑_{m_{K}=1}^{N+1−K} \binom{N − m_{K} + K − 2}{2K − 3} + ∑_{m_{1}+···+m_{K}=N−1} m_{1}m_{2} ⋯ m_{K}

= ∑_{m_{K}=2K−2}^{N+K−2} \binom{m_{K} − 1}{2K − 3} + \binom{N + K − 2}{2K − 1}

= \binom{N + K − 2}{2K − 2} + \binom{N + K − 2}{2K − 1} = \binom{N + K − 1}{2K − 1}.

Note that here we again used the identity

∑_{m=k}^{n−1} \binom{m − 1}{k − 1} = \binom{n − 1}{k}.

Proof of Theorem 5. We obtain

exp (X ∑_{n=1}^{∞} \frac{n^{2}}{n} q^{n}) = 1 + ∑_{k=1}^{∞} \frac{1}{k!}X^{k} (∑_{n=1}^{∞} nq^{n})^{k}
= 1 + ∑_{k=1}^{∞} \frac{1}{k!}X^{k} ( ∑_{m_{1}=1}^{∞} ⋯ ∑_{m_{k}=1}^{∞} m_{1} ⋯ m_{k}q^{m_{1}+···+m_{k}})
= 1 + ∑_{n=1}^{∞} ∑_{k=1}^{n} \frac{1}{k!}X^{k} ( ∑_{m_{1}+···+m_{k}=n} m_{1}m_{2} ⋯ m_{k}) q^{n}.

In the next step we apply Lemma 9 and obtain

1 + ∑_{n=1}^{∞} ∑_{k=1}^{n} \frac{1}{k!}\binom{n + k − 1}{2k − 1}X^{k}q^{n}

= 1 + ∑_{n=1}^{∞} \frac{X}{n!} ∑_{k=0}^{n−1} \frac{n!}{(k + 1)!}\binom{n + k}{2k + 1}X^{k}q^{n},

which yields the desired result.

5 Data

Figure 2: Quotient P_{n}^{σ} (1) /P_{n}^{φ_{1}} (1) depending on n.

\begin{matrix}n & P_{n}^{1} (1) & P_{n}^{σ} (1) & P_{n}^{2} (1) \\ 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 2 & 1.5 & 2 & 2.5 \\ 3 & 2.16667 & 3 & 5.16667 \\ 4 & 3.04167 & 5 & 10.0417 \\ 5 & 4.175 & 7 & 18.8417 \\ 6 & 5.62639 & 11 & 34.4181 \\ 7 & 7.46687 & 15 & 61.4752 \\ 8 & 9.78058 & 22 & 107.694 \\ 9 & 12.6669 & 30 & 185.485 \\ 10 & 16.2426 & 42 & 314.694 \\ 11 & 20.6448 & 56 & 526.768 \\ 12 & 26.0337 & 77 & 871.113 \\ 13 & 32.5961 & 101 & 1424.73 \\ 14 & 40.5493 & 135 & 2306.78 \\ 15 & 50.1454 & 176 & 3700.32 \\ 16 & 61.676 & 231 & 5884.91 \\ 17 & 75.4781 & 297 & 9284.78 \\ 18 & 91.9399 & 385 & 14540.1 \\ 19 & 111.508 & 490 & 22612 \\ 20 & 134.694 & 627 & 34935.4 \\ 21 & 162.087 & 792 & 53643.4 \\ 22 & 194.356 & 1002 & 81891.7 \\ 23 & 232.27 & 1255 & 124329 \\ 24 & 276.702 & 1575 & 187773 \\ 25 & 328.648 & 1958 & 282189 \end{matrix}

Table 1: Values of P_{n}^{φ_{1}} (1), the partition numbers P_{n}^{σ} (1), and P_{n}^{φ_{2}} (1).

References

[1] G. E. Andrews, The Theory of Partitions, Cambridge Univ. Press, 1998.

[2] B. Heim, F. Luca, and M. Neuhauser, On cyclotomic factors of polynomials related to modular forms, Ramanujan J. (2018). Available at http://dx.doi.org/10.1007/

s11139-017-9980-8.

[3] B. Heim, M. Neuhauser, and F. Rupp, Fourier coefficients of powers of the Dedekind eta function, Ramanujan J. (2017). Available at http://dx.doi.org/10.

1007/s11139-017-9923-4.

[4] D. Lehmer, The vanishing of Ramanujan’s τ(n), Duke Math. J. 14 (1947), 429–433.

[5] M. Newman, An identity for the coefficients of certain modular forms, J. Lond. Math. Soc. (2) 30 (1955), 488–493.

[6] K. Ono and S. Robins, Superlacunary cusp forms, Proc. Amer. Math. Soc. 123 (1995), 1021–1029.

[7] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, CBMS Regional Conference Series in Mathematics, Vol. 102, American Mathematical Society, 2004.

[8] J. Serre, Sur la lacunarité des puissances de η, Glasgow Math. J. 27 (1985), 203–221.

[9] H. S. Wilf, Generatingfunctionology, 3rd edition. A. K. Peters, 2006.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 05A10, 11B73, 11F20. Keywords: binomial coefficient, Dedekind eta function, Lehmer conjecture, linear recurrence, log-concavity, special sequence.

(Concerned with sequence \underline{A000041}, \underline{A000594}, \underline{A089231}, and \underline{A322970}.)

Received March 21 2018; revised version received January 3 2019. Published in Journal of Integer Sequences, January 6 2019.

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