A051784 Apply the "Stirling-Bernoulli transform" to A000081 = (1,1,1,2,4,9,20,...), rooted trees.

1, 0, 0, -6, 12, -270, 1500, -43806, 302652, -12857550, 132059100, -5733723006, 89592628092, -3922345875630, 79865827177500, -3844579915776606, 95745315867430332, -4957995149918778510, 151156611852387524700, -8193660691162420044606, 298062602379028314213372

Offset: 0

Author

N. J. A. Sloane, Dec 09 1999

Keywords

• sign

Comments

The "Stirling-Bernoulli transform" maps a sequence b_0, b_1, b_2, ... to a sequence c_0, c_1, c_2, ..., where if B has o.g.f. B(x), c has e.g.f. exp(x)*B(1-exp(x)).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Programs

Maple

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<3, 1,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= n-> add((-1)^k *k! *Stirling2(n+1, k+1)*b(k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 17 2013

Mathematica

b[n_] := b[n] = Module[{d, j}, If[n < 3, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]]; a[n_] := Sum[(-1)^k*k!*StirlingS2[n+1, k+1]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 01 2014, after Alois P. Heinz *)