A327731 Expansion of Product_{i>=1, j>=1} (1 + x^(i*(2*j - 1))).

1, 1, 1, 3, 3, 5, 8, 10, 14, 21, 28, 36, 51, 65, 86, 117, 148, 190, 251, 316, 402, 519, 647, 814, 1032, 1282, 1593, 1994, 2457, 3029, 3754, 4591, 5617, 6895, 8381, 10193, 12411, 14999, 18125, 21919, 26359, 31672, 38074, 45556, 54468, 65134, 77576, 92322

Offset: 0

Author

Ilya Gutkovskiy, Sep 23 2019

Keywords

• nonn

Comments

Weigh transform of A001227.

Crossrefs

Cf. A001227, A107742, A301800, A320650.

Programs

Mathematica

nmax = 47; CoefficientList[Series[Product[(1 + x^k)^DivisorSum[k, Mod[#, 2] &], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSum[d, Mod[#, 2] &], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

PARI

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^numdiv(k>>valuation(k, 2))))} \\ Andrew Howroyd, Sep 23 2019

Formula

G.f.: Product_{k>=1} (1 + x^k)^A001227(k).