A329163 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(j*(2*k - 1))).

1, 1, 3, 9, 22, 59, 156, 405, 1061, 2786, 7284, 19071, 49948, 130738, 342288, 896175, 2346134, 6142287, 16080852, 42100020, 110219366, 288558380, 755455128, 1977807393, 5177967900, 13556094631, 35490316938, 92914858431, 243254253904, 636847905903, 1667289469704, 4365020491362

Offset: 0

Author

Ilya Gutkovskiy, Nov 06 2019

Keywords

• nonn

Comments

Weigh transform of A032198.

Crossrefs

Cf. A032198, A102186, A329156.

Programs

Mathematica

nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(j (2 k - 1)), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Product[1/(1 - x^(2 k - 1)/(1 - x^(2 k - 1))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1 / (1 - x^(2*k - 1) / (1 - x^(2*k - 1))^2).

G.f.: Product_{k>=1} (1 + x^k)^A032198(k).

a(n) ~ c * phi^(2*n) / sqrt(5), where c = Product_{k>=2} 1/(1 - phi^(2 - 4*k)/(phi^(2 - 4*k) - 1)^2) = 1.07705428718361459418304978675229012... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 07 2019