A329467 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j)) * (1 + x^(2*i*j)).

1, 1, 3, 5, 10, 16, 31, 47, 81, 126, 204, 308, 487, 720, 1098, 1613, 2395, 3461, 5061, 7213, 10362, 14633, 20712, 28926, 40497, 56000, 77527, 106349, 145791, 198339, 269678, 364106, 491125, 658708, 882077, 1175392, 1563884, 2071363, 2739095, 3608040, 4744058, 6216087

Offset: 0

Author

Ilya Gutkovskiy, Nov 13 2019

Keywords

• nonn

Comments

Weigh transform of A069735.

Crossrefs

Cf. A000005, A001935, A006171, A069735, A107742, A320245.

Programs

Mathematica

nmax = 41; CoefficientList[Series[Product[((1 - x^(4 k))/(1 - x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d If[EvenQ[d], DivisorSigma[0, d] + DivisorSigma[0, d/2], DivisorSigma[0, d]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]

Formula

G.f.: Product_{i>=1, j>=1} (1 + x^(2*i*j)) / (1 - x^(i*(2*j - 1))).

G.f.: Product_{k>=1} ((1 - x^(4*k)) / (1 - x^k))^A000005(k).

G.f.: Product_{k>=1} (1 + x^k)^A069735(k).