A309240 - OEIS

%I #8 Jul 17 2019 16:38:37

%S 1,1,2,1,1,1,3,3,4,2,4,4,7,5,7,6,11,9,13,10,17,14,20,15,25,22,32,24,

%T 36,31,48,38,55,45,68,55,79,65,97,79,112,91,136,113,159,128,186,156,

%U 221,179,256,213,301,245,347,290,409,334,466,388,547,451,624,517,724,600,828,687,955,793,1088

%N Expansion of 1/((1 - x)*(1 - x^2)*(1 + x^3)*(1 + x^4)*(1 - x^5)*(1 - x^6)*(1 + x^7)*(1 + x^8)*...).

%H Vaclav Kotesovec, <a href="/A309240/b309240.txt">Table of n, a(n) for n = 0..10000</a>

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

%F G.f.: Product_{k>=1} (1 + x^(4*k-2)) / ((1 + x^(4*k-1)) * (1 - x^(4*k-3))).

%F a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n/6)) / (4 * 6^(1/8) * Pi^(3/4) * n^(5/8)). - _Vaclav Kotesovec_, Jul 17 2019

%t nmax = 70; CoefficientList[Series[Product[1/(1 + (-1)^(k (k + 1)/2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k - 2))/((1 + x^(4 k - 1)) (1 - x^(4 k - 3))), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 70; CoefficientList[Series[2/(QPochhammer[-1, -x^2] QPochhammer[x, -x^2]), {x, 0, nmax}], x]

%Y Cf. A000700, A035451, A147599, A300574.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 17 2019