0,5

For the o.g.f.s of diagonal k of the Stirling2 triangle one has a similar result. See A008517 (second-order Eulerian triangle).

A(m,x), the o.g.f. for column m satisfies the recurrence A(m,x)= x*(x*diff(A(m-1,x),x) + m*A(m-1,x))/(1-(m+1)*x), for m>=1, and A(0,x)=1/(1-x).

The column sequences start with A000012 (powers of 1), A000295 (Eulerian numbers), A112495-A112497.

The e.g.f. for the sequence in column k+1, k>=0, of A008278, i.e. for the diagonal k>=0 of the Stirling2 triangle A048993, is exp(x)*sum(a(k,m)*(x^(m+k))/(m+k)!,m=0..k).

W. Lang, First ten rows.

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

[1];[1,1];[1,4,3];[1,11,25,15];[1,26,130,210,105];...

The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).

Row sums give A006351(k+1), k>=0.

Sequence in context: A109692 A128813 A109062 this_sequence A010305 A098234 A128320

Adjacent sequences: A112490 A112491 A112492 this_sequence A112494 A112495 A112496

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 14 2005