A284092 Number of partitions of n into distinct parts 8k+1 or 8k+7.

1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 3, 5, 5, 3, 1, 0, 0, 2, 5, 7, 7, 5, 2, 0, 1, 3, 7, 11, 11, 7, 3, 1, 1, 5, 11, 15, 15, 11, 5, 1, 2, 7, 15, 22, 22, 15, 7, 2, 3, 11, 22, 30, 30, 22, 11, 4, 5, 15, 30, 42, 42

Offset: 0

Author

Seiichi Manyama, Mar 20 2017

Keywords

• nonn

Comments

Convolution of A284093 and A284095.

Links

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

Crossrefs

Cf. A035453, A284093, A284095.

Cf. Product_{k>0} (1 + x^(m*k - 1)) * (1 + x^(m*k - m + 1)): A003105 (m=3), A000700 (m=4), A203776 (m=5), A098884 (m=6), A281459 (m=7), this sequence (m=8).

Programs

Mathematica

CoefficientList[Series[Product[(1 + x^(8*k - 1)) * (1 + x^(8*k - 7)) , {k, 1, 81}], {x, 0, 81}], x] (* Indranil Ghosh, Mar 20 2017 *)

PARI

Vec(prod(k=1, 81, (1 + x^(8*k - 1)) * (1 + x^(8*k - 7))) + O(x^82)) \\ Indranil Ghosh, Mar 20 2017

Formula

G.f.: Product_{k>0} (1 + x^(8*k - 1)) * (1 + x^(8*k - 7)).

a(n) ~ exp(sqrt(n/3)*Pi/2) / (4*3^(1/4)*n^(3/4)) * (1 + (11*Pi/(192*sqrt(3)) - 3*sqrt(3)/(4*Pi))/sqrt(n)). - Vaclav Kotesovec, Mar 20 2017