A035440 Number of partitions of n into parts 7k+5 or 7k+6.

0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 2, 2, 1, 2, 3, 5, 5, 4, 3, 3, 5, 8, 8, 8, 6, 6, 8, 12, 13, 14, 13, 12, 13, 17, 20, 23, 23, 22, 21, 26, 30, 36, 38, 38, 37, 40, 45, 53, 59, 62, 61, 65, 67, 78, 88, 96, 100, 102, 104, 114, 128, 144, 152, 160, 160, 170, 186, 208

Offset

1,12

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 26 2015

Links

Robert Israel, Table of n, a(n) for n = 1..3000

Formula

a(n) ~ exp(2*Pi*sqrt(n/21)) * Gamma(5/7) * Gamma(6/7) * 7^(1/28) / (4 * 3^(15/28) * Pi^(3/7) * n^(29/28)). - Vaclav Kotesovec, Aug 26 2015

G.f.: Product_{k>=0} 1/((1-q^(7*k+5)) * (1-q^(7*k+6)) ). - Joerg Arndt, Jun 22 2020

Maple

N:= 100:
kmax:= floor((N-5)/7);
g:= mul(1/(1-x^(7*k+5))/(1-x^(7*k+6)), k=0..kmax):
S:= series(g, x, N+1):
seq(coeff(S, x, n), n=1..N); # Robert Israel, Jun 22 2020

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(7k+5))*(1 - x^(7k+6))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 26 2015 *)

Crossrefs

Sequence in context: A276719 A276837 A269941 * A029878 A182458 A238093

Adjacent sequences: A035437 A035438 A035439 * A035441 A035442 A035443

Keyword

nonn

Author

Olivier Gérard

Status

approved