0,4

Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos Mar 04 2004

Occurs also as first column of a matrix-inversion occuring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007

Gottfried Helms, Discussion of a problem concerning summing of like powers

E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))). a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).

a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 17 2005

Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]

(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(y/(1-y^2), y, exp(x+x*O(x^n))-1), n))

Cf. A052841.

Sequence in context: A096965 A159597 A100309 this_sequence A075996 A093168 A097493

Adjacent sequences: A089674 A089675 A089676 this_sequence A089678 A089679 A089680

easy,nonn

Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004