RFE Feb 2026: Generalized Stirling numbers of the first kind

[Sean A. Irvine]

Hi,

For this month I would like help with a naming issue.

There are 17 sequences with the exact same name: Generalized Stirling number triangle of [the] first kind (A048176, A049444, A049458, A049459, A049460, A051142, A051150, A051151, A051186, A051187, A051231, A051338, A051339, A051379, A051380, A051523, A094646, and possibly other related sequences with similar names). I would like someone with knowledge of this topic to come up with improved and distinct names for these sequences.

(If you don't have, or don't want to consume your edit slots, then proposed names can be posted here, or sent to me directly.)

Thanks to Martin Fuller who found the g.f.'s needed for my January question and the contributors who subsequently improved the corresponding sequences.

I track these requests for enhancement here:

https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements

Sean.

[Natalia L. Skirrow]

they are respectively (where [n,k] = |Stirling1(n,k)| = Stirling1(n,k)*(-1)^(n-k) and [n,k]_r the r-Stirling numbers)

A051142(n,k) = [n,k]*(-4)^(n-k).

A051150(n,k) = [n,k]*(-5)^(n-k).

A051151(n,k) = [n,k]*(-6)^(n-k).

A051186(n,k) = [n,k]*(-7)^(n-k).

A051187(n,k) = [n,k]*(-8)^(n-k).

A051231(n,k) = [n,k]*(-9)^(n-k).

A048176(n,k) = [n,k]*(-10)^(n-k).

A049444(n,k) = [n+2,k+2]_2 * (-1)^(n-k).

A049458(n,k) = [n+3,k+3]_3 * (-1)^(n-k).

A049459(n,k) = [n+4,k+4]_4 * (-1)^(n-k).

A049460(n,k) = [n+5,k+5]_5 * (-1)^(n-k).

A051338(n,k) = [n+6,k+6]_6 * (-1)^(n-k).

A051339(n,k) = [n+7,k+7]_7 * (-1)^(n-k).

A051379(n,k) = [n+8,k+8]_8 * (-1)^(n-k).

A051380(n,k) = [n+9,k+9]_9 * (-1)^(n-k).

A051523(n,k) = [n+10,k+10]_10 * (-1)^(n-k).

and finally,

   A094646(n,k)

= [n-2,k-2]_-2

= [3,3-n+k]_{3-n} * (-1)^(n-k)

= {2-k,2-n}_3

= {n-k-3,-3}_{n-2} * (-1)^(n-k)

due to the fourfold reflection identities for r-Stirling numbers (see this section of one of my pages); none of these have a direct combinatorial interpretation, so I don't think there is anything more meaningful than the existing description

T(n,k) = 2*abs(S1(n-2,k)) - 3*abs(S1(n-2,k-1)) + abs(S1(n-2,k-2)), n >= 2, with S1(n,k) = Stirling1(n,k) = A048994(n,k).

to mention

[Thomas Scheuerle]

All these sequences have in common that row n of the triangle can be interpreted as coefficients of a polynomial with real roots, and row n has one root added to the polynomial of row n-1. 

For example the polynomial P(x)_n  from row n in A049444 can be described as (n+1)!*Product_{k=1,n)(x-1/(1+k)).

Would it make sense to find names based on these polynomials?

[Md. Rad Sarar Anando]

In my opinion, these sequences should be grouped together in such a way that anyone interested in any of the sequences here can view them all at once. Perhaps they could all be related.

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[Natalia L. Skirrow]

yes, that is what Andrei Z. Broder does with the r-Stirling numbers

[Sean A. Irvine]

I agree. We should try to do that in multiple ways, first by retaining a certain common structure to the names, and second by cross-references between the sequences.

A051141 which also belongs to this series might serve as a template for the names.

Sean.

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