A292951 Expansion of e.g.f. exp(x^2 * (1 - exp(x))).

1, 0, 0, -6, -12, -20, 330, 2478, 11704, -15192, -751050, -7817150, -38408172, 151402524, 5793891922, 69046056870, 393083614320, -2517944476592, -98819987200146, -1384209703077750, -9376308260215220, 67288368700200900, 3186749671049174538

Offset: 0

Author

Seiichi Manyama, Sep 27 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..200

Crossrefs

Column k=2 of A292894.

Cf. A240989.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x^2 (1-Exp[x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 16 2021 *)

PARI

x='x+O('x^66); Vec(serlaplace(exp(x^2*(1-exp(x)))))

PARI

a(n) = n!*sum(k=0, n\3, (-1)^k*stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)

a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * Stirling2(n-2*k,k)/(n-2*k)!.

a(0) = 1; a(n) = -(n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)