A369752 Expansion of e.g.f. exp(1 - (1+x)^4).

1, -4, 4, 56, -104, -2464, 1696, 181184, 462016, -17069824, -141580544, 1593913856, 33015560704, -47193585664, -6973651011584, -50207289585664, 1214484253413376, 25500259291480064, -72069247145590784, -8696105637665603584, -81680899029758541824

Offset: 0

Author

Seiichi Manyama, Jan 30 2024

Keywords

• sign

Crossrefs

Column k=4 of A369738.

Cf. A000587, A202824.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Exp[1-(1+x)^4],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 29 2024 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1-(1+x)^4)))

Formula

a(0) = 1; a(n) = -4 * (n-1)! * Sum_{k=1..min(4,n)} binomial(3,k-1) * a(n-k)/(n-k)!.

a(n) = Sum_{k=0..n} 4^k * Stirling1(n,k) * A000587(k).

D-finite with recurrence a(n) +4*a(n-1) +12*(n-1)*a(n-2) +12*(n-1)*(n-2)*a(n-3) +4*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 02 2024