0,2

This is the row reversed second-order Eulerian triangle A008517(k+1,k+1-m). For references see A008517.

The o.g.f. for the k-th diagonal, k>=1, of the unsigned Stirling1 triangle |A008275| is G1(1,x)=1/(1-x) if k=1, and G1(k,x)= g1(k-2,x)/(1-x)^(2*k-1), if k>=2, with the row polynomials g1(k;x):=sum(a(k,m)*x^m,m=0..k).

The recurrence eq. for the row polynomials is: g1(k,x)=((k+1)+k*x))*g1(k-1,x) + x*(1-x)*diff(g1(k-1,x),x), k>=1, with input g1(0,x):=1.

The column sequences start with A000142 (factorials), A002538, A002539, A112008, A112485.

This o.g.f. 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 have been rediscussed.

W. Lang, First 10 rows.

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

[1]; [2,1]; [6,8,1]; [24,58,22,1]; [120,444,328,52,1]; ...

G.f. for k=3 sequence A000914(n-1), [2,11,35,85,175,322,546,...], is G1(3,x)= g1(1,x)/(1-x)^5= (2+x)/(1-x)^5.

Row sums give A001147(k+1)= (2*k+1)!!, k>=0.

Adjacent sequences: A112004 A112005 A112006 this_sequence A112008 A112009 A112010

Sequence in context: A059364 A047708 A110608 this_sequence A113374 A136470 A138510

nonn,easy,tabl

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