A344097 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,4).

1, 5, 25, 120, 505, 2027, 7740, 28345, 100355, 344815, 1154130, 3773955, 12085125, 37971645, 117258755, 356386016, 1067364240, 3153415695, 9198749905, 26516197720, 75586609016, 213212467695, 595482274750, 1647568369230, 4517987288720, 12284672226583, 33133931688645

Offset: 0

Author

Ilya Gutkovskiy, May 09 2021

Keywords

• nonn

Crossrefs

Cf. A000332, A174002, A219555, A255052, A338645, A343200, A344098.

Programs

Mathematica

nmax = 26; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 4, 4], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 4, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A174002(d) ) * a(n-k).