A376624 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

1, 1, 2, 4, 6, 8, 13, 18, 23, 33, 44, 57, 77, 99, 125, 163, 207, 259, 328, 407, 503, 626, 769, 938, 1149, 1397, 1687, 2044, 2458, 2943, 3531, 4213, 5011, 5957, 7055, 8334, 9838, 11580, 13594, 15948, 18661, 21790, 25425, 29593, 34386, 39918, 46250, 53501, 61824, 71325

Offset: 0

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A053282, A072223, A376542, A376622, A376581, A376658.

Programs

Mathematica

nmax=60; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j-1))^2.

a(n) ~ (r^(3/4)/sqrt(8*(1 + 3*r^2))) * A376658^sqrt(n) / sqrt(n), where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.