login
Expansion of (1 - x^2)*Product_{k>=2} (1 + x^k)^k.
0

%I #6 May 04 2018 06:54:57

%S 1,0,1,3,3,8,12,21,34,59,93,150,242,377,595,922,1419,2171,3310,4988,

%T 7507,11218,16674,24676,36353,53295,77828,113209,163989,236736,340517,

%U 488108,697407,993350,1410455,1996968,2819280,3969260,5573541,7806141,10905640,15199138,21133212

%N Expansion of (1 - x^2)*Product_{k>=2} (1 + x^k)^k.

%C First differences of A026007.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F G.f.: (1 - x)*exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)).

%F a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * Zeta(3)^(1/2) / (2^(13/12) * sqrt(Pi) * n). - _Vaclav Kotesovec_, May 04 2018

%t nmax = 42; CoefficientList[Series[(1 - x^2) Product[(1 + x^k)^k, {k, 2, nmax}], {x, 0, nmax}], x]

%t nmax = 42; CoefficientList[Series[(1 - x) Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

%Y Cf. A002865, A026007, A087897, A191659, A298598, A302832.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, May 02 2018