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Expansion of Product_{k>=1} (1 + x^(8*k-1)).
2

%I #16 Nov 24 2020 07:49:34

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

%T 0,0,0,0,2,1,0,0,0,0,0,1,2,1,0,0,0,0,0,1,3,1,0,0,0,0,0,2,3,1,0,0,0,0,

%U 0,3,4,1,0,0,0,0,1,4,4,1,0,0,0,0,1,5,5,1

%N Expansion of Product_{k>=1} (1 + x^(8*k-1)).

%C Number of partitions into distinct parts 8*k-1.

%H Vaclav Kotesovec, <a href="/A284093/b284093.txt">Table of n, a(n) for n = 0..20000</a>

%F a(n) ~ exp(sqrt(n/6)*Pi/2) / (2^(21/8) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(384*sqrt(6)) - 3*sqrt(3/2)/(2*Pi))/sqrt(n)). - _Vaclav Kotesovec_, Mar 20 2017

%F G.f.: Sum_{k>=0} x^(k*(4*k + 3)) / Product_{j=1..k} (1 - x^(8*j)). - _Ilya Gutkovskiy_, Nov 24 2020

%t CoefficientList[Series[Product[(1 + x^(8*k - 1)) , {k, 1, 91}], {x, 0, 91}], x] (* _Indranil Ghosh_, Mar 20 2017 *)

%t nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 8] == 7, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Mar 20 2017 *)

%o (PARI) Vec(prod(k=1, 91, (1 + x^(8*k - 1))) + O(x^92)) \\ _Indranil Ghosh_, Mar 20 2017

%Y Cf. Product_{k>=1} (1 + x^(m*k-1)): A262928 (m=3), A147599 (m=4), A281243 (m=5), A281244 (m=6), A281245 (m=7), this sequence (m=8).

%K nonn

%O 0,39

%A _Seiichi Manyama_, Mar 20 2017