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A360784
Number of multisets of nonempty strict integer partitions with a total of n parts and total sum of 2n.
3
1, 1, 3, 8, 18, 39, 86, 175, 352, 688, 1318, 2472, 4576, 8322, 14959, 26560, 46657, 81130, 139866, 239047, 405496, 682891, 1142466, 1899344, 3139432, 5160455, 8438871, 13732292, 22242647, 35867937, 57597730, 92121145, 146775205, 232998683, 368579188, 581091003
OFFSET
0,3
LINKS
FORMULA
a(n) = A360763(2n,n).
EXAMPLE
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]}.
MAPLE
h:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..35);
MATHEMATICA
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]]]]];
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}]]]];
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}]]]];
a[n_] := Coefficient[b[2 n, 2 n], x, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 21 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 20 2023
STATUS
approved