A035468 Number of partitions of n into parts 8k+6 or 8k+7.

0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 5, 4, 4, 2, 2, 3, 5, 7, 8, 7, 5, 4, 5, 7, 12, 12, 13, 11, 9, 8, 12, 16, 19, 21, 20, 17, 16, 17, 24, 28, 33, 33, 33, 28, 29, 34, 41, 48, 54, 53, 50, 48, 52, 58, 72, 79, 85, 83, 81, 80, 88, 100, 116, 127

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

In general, if a > 0, b > 0, c > 0, d > 0, gcd(a,b) = 1, gcd(c,d) = 1 and g.f. = Product_{k>=0} 1/((1 - x^(a*k+b)) * (1 - x^(c*k+d))), then a(n) ~ Gamma(b/a) * Gamma(d/c) * a^((2*b/a - 2*d/c - 1)/4) * c^((2*d/c - 2*b/a - 1)/4) * (a+c)^((2*b/a + 2*d/c - 1)/4) * Pi^(b/a + d/c - 2) * exp(Pi*sqrt(2*(1/a + 1/c)*n/3)) / (2^((2*b/a + 2*d/c + 7)/4) * 3^((2*b/a + 2*d/c - 1)/4) * n^((1 + 2*b/a + 2*d/c)/4)). - Vaclav Kotesovec, Aug 27 2015

Links

• Robert Price, Table of n, a(n) for n = 1..1000

Programs

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+6))*(1 - x^(8k+7))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 27 2015 *)
nmax = 60; kmax = nmax/8;
s = Flatten[{Range[0, kmax]*8 + 6}~Join~{Range[0, kmax]*8 + 7}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* Robert Price, Aug 04 2020 *)

Formula

a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(3/4) * Gamma(7/8) / (2^(29/16) * 3^(9/16) * Pi^(3/8) * n^(17/16)). - Vaclav Kotesovec, Aug 27 2015