0,4

The k-th diagonal of |A008275| appears as k-th column in |A008276| with k-1 leading zeros.

The recurrence, given below, is derived from diff(g1(k,x),x) - g1(k,x)= x*diff(g1(k-1,x),x) + g1(k-1,x), k>=1, with input g(-1,x):=0 and initial condition g1(k,0)=1, k>=0. This differential recurrence for the e.g.f. g1(k,x) follows from the one for unsigned Stirling1 numbers.

The column sequences start with A000142 (factorials), A001705, A112487- A112491, for m=0,...,5.

The main diagonal gives (2*k-1)!! = A001147(k), k>=1.

This computation was inspired by the preprint arXiv:math-ph/0509008 v1 5 Sep 2005 by C. M. Bender, D. C. Brody and B. K. Meister: "Bernoulli-like polynomials associated with Stirling Numbers", where the Stirling polynomials are discussed.

The e.g.f. for the k-th diagonal, k>=1, of the unsigned Stirling1 triangle |A008275| with k-1 leading zeros is g1(k-1,x)=exp(x)*sum(a(k,m)*(x^(k-1+m))/(k-1+m)!,m=0..k-1).

W. Lang, First 10 rows.

a(k, m)= (k+m)*a(k-1, m)+(k+m-1)*a(k-1, m-1) for m>=k>=0, a(0, 0)=1, a(k, -1):=0, a(k, m)=0 if k<m.

[1]; [1,1]; [2,5,3]; [6,26,35,15]; [24,154,340,315,105]; ...

k=3 column of |A008276| is [0,0,2,11,35,85,175,...] (see A000914), its e.g.f. exp(x)*(2*x^2/2! + 5* x^3/3! + 3*x^4/4!).

Cf. A112007 (triangle for o.g.f.s for unsigned Stirling1 diagonals).

Adjacent sequences: A112483 A112484 A112485 this_sequence A112487 A112488 A112489

Sequence in context: A037852 A024871 A111202 this_sequence A078383 A125512 A135587

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Sep 12 2005