A051784 - OEIS

%I #15 Jun 27 2016 17:38:01

%S 1,0,0,-6,12,-270,1500,-43806,302652,-12857550,132059100,-5733723006,

%T 89592628092,-3922345875630,79865827177500,-3844579915776606,

%U 95745315867430332,-4957995149918778510,151156611852387524700,-8193660691162420044606,298062602379028314213372

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

%C 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)).

%H Alois P. Heinz, <a href="/A051784/b051784.txt">Table of n, a(n) for n = 0..200</a>

%p with(numtheory):

%p b:= proc(n) option remember; local d, j; `if` (n<3, 1,

%p       (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))

%p     end:

%p a:= n-> add((-1)^k *k! *Stirling2(n+1, k+1)*b(k), k=0..n):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, May 17 2013

%t 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_ *)

%K sign

%O 0,4

%A _N. J. A. Sloane_, Dec 09 1999