A284095 - OEIS

%I #18 Nov 24 2020 10:24:11

%S 1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,2,1,0,0,0,0,0,1,

%T 2,1,0,0,0,0,0,1,3,2,0,0,0,0,0,1,3,3,1,0,0,0,0,1,4,4,1,0,0,0,0,1,4,5,

%U 2,0,0,0,0,1,5,7,3,0,0,0,0,1,5,8,5,1,0,0,0

%N Expansion of Product_{k>=0} (1 + x^(8*k+1)).

%C Number of partitions of n into distinct parts congruent to 1 mod 8.

%H Vaclav Kotesovec, <a href="/A284095/b284095.txt">Table of n, a(n) for n = 0..20000</a>

%F a(n) ~ exp(sqrt(n/6)*Pi/2) / (2^(15/8) * 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, 0, 91}], {x, 0, 91}], x] (* _Indranil Ghosh_, Mar 20 2017 *)

%t nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 8] == 1, 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=0, 91, (1 + x^(8*k + 1))) + O(x^92)) \\ _Indranil Ghosh_, Mar 20 2017

%Y Cf. Product_{k>=0} (1 + x^(m*k+1)): A261612 (m=3), A169975 (m=4), A280454 (m=5), A280456 (m=6), A280457 (m=7), this sequence (m=8).

%K nonn

%O 0,27

%A _Seiichi Manyama_, Mar 20 2017