A109703 Number of partitions of n into parts each equal to 1 mod 7.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 31, 34, 38, 41, 43, 44, 45, 46, 50, 55, 60, 63, 65, 66, 68, 72, 79, 85, 90, 93, 95, 97, 103, 111, 120, 127, 132, 135

Offset: 0

Author

Erich Friedman, Aug 07 2005

Keywords

• nonn

Examples

			a(15)=3 because we have 15=8+1+1+1+1+1+1+1=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, Graph - The asymptotic ratio

Crossrefs

Cf. A284099.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), A109697 (m=5), A109701 (m=6), this sequence (m=7), A277090 (m=8).

Programs

Maple

g:=1/product(1-x^(1+7*j),j=0..20): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=0..77); # Emeric Deutsch, Apr 14 2006

Mathematica

nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+1)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

Formula

G.f.: 1/product(1-x^(1+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006

a(n) ~ Gamma(1/7) * exp(Pi*sqrt(2*n/21)) / (2^(11/7) * 3^(1/14) * 7^(3/7) * Pi^(6/7) * n^(4/7)) * (1 - (2*sqrt(6/7)/(7*Pi) + 13*Pi/(168*sqrt(42))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017

a(n) = (1/n)*Sum_{k=1..n} A284099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(7*j)). - Ilya Gutkovskiy, Jul 17 2019

Extensions

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015