A291748 - OEIS

%I #7 Aug 31 2017 06:22:52

%S 1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,

%T 0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,2,1,0,0,0,0,1,2,1,0,0,0,0,0,0,0,0,1,1,

%U 0,0,0,1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0

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

%H Vaclav Kotesovec, <a href="/A291748/b291748.txt">Table of n, a(n) for n = 0..100000</a>

%F a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3)/4) * ((sqrt(2)-1) * Zeta(3/2))^(1/3) / (2 * sqrt(6) * Pi^(1/3) * n^(5/6)).

%t nmax = 200; CoefficientList[Series[Product[(1 + x^(2*k^2 - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - 2*k^2 + 2]], {j, nmax, 2*k^2 - 1, -1}];, {k, 2, Sqrt[(nmax + 1)/2] + 1}]; poly

%Y Cf. A033461, A291749.

%K nonn

%O 0,50

%A _Vaclav Kotesovec_, Aug 31 2017