A300413 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 - x^prime(k)).

1, 0, 2, 2, 2, 6, 4, 10, 10, 14, 20, 22, 32, 38, 48, 60, 74, 90, 112, 134, 164, 196, 236, 282, 336, 398, 472, 554, 652, 766, 890, 1046, 1206, 1408, 1624, 1876, 2168, 2486, 2860, 3276, 3744, 4282, 4878, 5554, 6316, 7160, 8124, 9186, 10388, 11722, 13216, 14876, 16732, 18794, 21084, 23636

Offset: 0

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

• nonn

Comments

Convolution of the sequences A000586 and A000607.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000040, A000586, A000607, A015128, A080054, A103265, A280263, A280366, A300414, A300415.

Programs

Mathematica

nmax = 55; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A000040(k))/(1 - x^A000040(k)).

log(a(n)) ~ Pi*sqrt(2*n/log(n/3)) * (1/3 + 2*sqrt(log(n/3) / log(2*n/3)) / 3). - Vaclav Kotesovec, Jan 12 2021