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A360784 Number of multisets of nonempty strict integer partitions with a total of n parts and total sum of 2n.

%I A360784 #15 Nov 21 2023 08:39:34
%S A360784 1,1,3,8,18,39,86,175,352,688,1318,2472,4576,8322,14959,26560,46657,
%T A360784 81130,139866,239047,405496,682891,1142466,1899344,3139432,5160455,
%U A360784 8438871,13732292,22242647,35867937,57597730,92121145,146775205,232998683,368579188,581091003
%N A360784 Number of multisets of nonempty strict integer partitions with a total of n parts and total sum of 2n.
%H A360784 Alois P. Heinz, <a href="/A360784/b360784.txt">Table of n, a(n) for n = 0..450</a>
%F A360784 a(n) = A360763(2n,n).
%e A360784 a(3) = 8: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[1],[4]}, {[1],[2],[3]}, {[2],[2],[2]}.
%p A360784 h:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A360784       `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
%p A360784     end:
%p A360784 g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p A360784       g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
%p A360784     end:
%p A360784 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A360784      `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
%p A360784     end:
%p A360784 a:= n-> coeff(b(2*n$2), x, n):
%p A360784 seq(a(n), n=0..35);
%t A360784 h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]];
%t A360784 g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i] + k - 1, k], {k, 0, j}]]]];
%t A360784 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
%t A360784 a[n_] := Coefficient[b[2 n, 2 n], x, n];
%t A360784 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Nov 21 2023, after _Alois P. Heinz_ *)
%Y A360784 Cf. A000009, A360763, A360785.
%K A360784 nonn
%O A360784 0,3
%A A360784 _Alois P. Heinz_, Feb 20 2023