A300413 - OEIS

%I #9 Jan 12 2021 10:47:49

%S 1,0,2,2,2,6,4,10,10,14,20,22,32,38,48,60,74,90,112,134,164,196,236,

%T 282,336,398,472,554,652,766,890,1046,1206,1408,1624,1876,2168,2486,

%U 2860,3276,3744,4282,4878,5554,6316,7160,8124,9186,10388,11722,13216,14876,16732,18794,21084,23636

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

%C Convolution of the sequences A000586 and A000607.

%H Vaclav Kotesovec, <a href="/A300413/b300413.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Par#part">Index entries for related partition-counting sequences</a>

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

%F 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

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

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 05 2018