%I #20 Nov 30 2023 12:02:12

%S 1,1,3,13,67,471,3591,33573,329043,3919387,47827093,663429603,

%T 9764977399,156308277139,2653548775671,48880554540093,934560430625523,

%U 19120475459863299,413057291727064689,9325822483756554831,221409989926026560757,5513060361601128645777

%N E.g.f.: Product_{n>=1} Sum_{k>=0} (x^k/k!)^n.

%H Alois P. Heinz, <a href="/A182666/b182666.txt">Table of n, a(n) for n = 0..444</a> (first 301 terms from Vaclav Kotesovec)

%H Kévin Perrot, Sylvain Sené, and Léah Tapin, <a href="https://arxiv.org/abs/2304.09664">On countings and enumerations of block-parallel automata networks</a>, arXiv:2304.09664 [cs.DM], 2023.

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 67*x^4/4! + 471*x^5/5! + 3591*x^6/6! +...

%e where

%e A(x) = [Sum_{k>=0} x^k/k!] * [Sum_{k>=0} (x^k/k!)^2] * [Sum_{k>=0} (x^k/k!)^3] * [Sum_{k>=0} (x^k/k!)^4] *...

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p b(n-i*j, i-1)*combinat[multinomial](n, n-i*j, j$i), j=0..n/i)))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Jul 18 2023

%t multinomial[n_, k_List] := n!/Times @@ (k!);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* multinomial[n, Join[{n - i*j}, Table[j, {i}]]], {j, 0, n/i}]]];

%t a[n_] := b[n, n];

%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Nov 30 2023, after _Alois P. Heinz_ *)

%o (PARI) {a(n)=n!*polcoeff(prod(m=1,n,sum(k=0,n\m+1,x^(m*k)/k!^m)+x*O(x^n)),n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A005651.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 23 2012