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(* 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 *)

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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

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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

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