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MATHEMATICA
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h[l_] := With[{n=Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, k_, l_] := g[n, i, l, k] = If[n == 0, h[l]^k, If[i < 1, 0, g[n, i - 1, k, l] + If[i > n, 0, g[n - i, i, k, Append[l, i]]]]];
a[n_] := If[n == 0, 1, g[n, n, 4, {}]];
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