A309085 a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.

1, -4, 12, -20, -20, 172, 108, -2388, -3220, 47532, 161900, -1062740, -8532628, 13623212, 431041132, 1206169260, -17833021588, -169685043796, 180187176044, 13462762665132, 79377664422252, -553096696140884, -11670986989785492, -44371854928405844, 829755609457185644

Offset: 0

Author

Ilya Gutkovskiy, Jul 11 2019

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..564

Crossrefs

Column k = 4 of A292861.

Cf. A000587, A078944, A213170, A309084, A318179.

Programs

Magma

[1] cat [(&+[((-4)^k*StirlingSecond(m,k)):k in [0..m]]):m in [1..24]]; // Marius A. Burtea, Jul 11 2019

Mathematica

Table[Exp[4] Sum[(-4)^k k^n/k!, {k, 0, Infinity}], {n, 0, 24}]
Table[BellB[n, -4], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Sum[(-4)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!

PARI

a(n) = sum(k=0, n, (-4)^k * stirling(n,k,2)); \\ Michel Marcus, Jul 12 2019

Formula

G.f.: Sum_{j>=0} (-4)^j*x^j / Product_{k=1..j} (1 - k*x).

E.g.f.: exp(4*(1 - exp(x))).

a(n) = Sum_{k=0..n} (-4)^k * Stirling2(n,k).