A309084 a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.

1, -3, 6, -3, -21, 24, 195, -111, -3072, -4053, 57003, 277854, -697539, -12261567, -29861778, 371727465, 3511027599, 2028432480, -188521156857, -1470389129931, 1655487186864, 121873222577823, 915525253963023, -2095901567014530, -103715912230195863, -836215492271268459

Offset: 0

Author

Ilya Gutkovskiy, Jul 11 2019

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..571

Crossrefs

Column k = 3 of A292861.

Cf. A000587, A027710, A213170, A309085, A317996.

Programs

Magma

[1] cat [(&+[((-3)^k*StirlingSecond(m, k)):k in [0..m]]):m in [1..25]]; // Marius A. Burtea, Jul 27 2019

Maple

b:= proc(n, m) option remember; `if`(n=0,
      (-3)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 17 2022

Mathematica

Table[Exp[3] Sum[(-3)^k k^n/k!, {k, 0, Infinity}], {n, 0, 25}]
Table[BellB[n, -3], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[(-3)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!

Formula

G.f.: Sum_{j>=0} (-3)^j*x^j / Product_{k=1..j} (1 - k*x).

E.g.f.: exp(3*(1 - exp(x))).

a(n) = Sum_{k=0..n} (-3)^k * Stirling2(n,k).