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Recurrent Combinatorial Sums and Binomial-Type Theorems
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Michael Z. Spivey

Department of Mathematics and Computer Science

University of Puget Sound

Tacoma, WA 98416-1043

USA

**Abstract:**

We prove that several known combinatorial sum identities, including
identities featuring the binomial coefficients, both kinds of Stirling
numbers, the Lah numbers, both kinds of associated Stirling numbers,
and first- and second-order Eulerian numbers, are all special cases of
a single identity satisfying a simple one-term recurrence. Moreover,
subject to a certain set of restrictions, these special cases are the
only identities featuring the aforementioned numbers that also satisfy
this one-term recurrence. We then derive formulas for the coefficients
involved when converting between multifactorial powers (a generalization
of rising and of falling factorial powers) and ordinary powers, as
well as some conditions under which numbers satisfying a general two-term
combinatorial recurrence can be considered inverses of each other. Next,
we derive conditions under which numbers that satisfy a general two-term
recurrence also satisfy a binomial-type theorem, and we show that the
binomial coefficients and the Stirling numbers of the first kind are the
only numbers mentioned earlier that exhibit these conditions. Finally,
we give a combinatorial proof of a binomial-type theorem satisfied by
the Stirling numbers of the first kind.

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(Concerned with sequences
A000142
A001147
A007318
A008275
A008277
A008292
A008297
A008299
A008306
A008517
A048994
A105278.)

Received May 9 2019; revised version received December 13 2019; January 9 2020.
Published in *Journal of Integer Sequences*,
June 3 2020.

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