proposed
approved
editing
(* Second program; *): *)
(* Second program; *)
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, x^k, b[n, i-1, k] + Function[t, k*b[n-i, t, k] + b[n-i, t, k + 1]][Min[n-i, i-1]]]];
T[n_] := PadRight[CoefficientList[b[n, n, 0], x], n + 1];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)
Cf. A000110, A008277, A050342, A060016, A072706, A270995, A271619, A279375, A279785, A326701, A330459, A330462, A330463, A330759.
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, x^k, b(n, i-1, k) +(t-> k*
b(n-i, t, k)+b(n-i, t, k+1))(min(n-i, i-1))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..15); # Alois P. Heinz, Dec 29 2019
1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 0, 0, 0, 0, 4, 5, 1, 0, 0, 0, 0, 5, 6, 1, 0, 0, 0, 0, 0, 6, 9, 2, 0, 0, 0, 0, 0, 0, 8, 13, 3, 0, 0, 0, 0, 0, 0, 0, 10, 23, 10, 1, 0, 0, 0, 0, 0, 0, 0, 12, 27, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 40, 19, 2, 0, 0, 0, 0, 0, 0, 0, 0
Andrew Howroyd, <a href="/A330460/b330460.txt">Table of n, a(n) for n = 0..1325</a> (rows n=0..50)
(PARI)
A(n)={my(v=Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n)))); vector(#v, n, my(p=v[n]); vector(n, k, sum(i=k, n, polcoef(p, i-1)*stirling(i-1, k-1, 2))))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019