A284095 Expansion of Product_{k>=0} (1 + x^(8*k+1)).

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

Offset: 0

Author

Seiichi Manyama, Mar 20 2017

Keywords

• nonn

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Crossrefs

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).

Programs

Mathematica

CoefficientList[Series[Product[(1 + x^(8*k + 1)) , {k, 0, 91}], {x, 0, 91}], x] (* Indranil Ghosh, Mar 20 2017 *)
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 *)

PARI

Vec(prod(k=0, 91, (1 + x^(8*k + 1))) + O(x^92)) \\ Indranil Ghosh, Mar 20 2017

Formula

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

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