A109708 Number of partitions of n into parts each equal to 6 mod 7.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 2, 1, 2, 3, 5, 5, 5, 2, 2, 3, 5, 6, 8, 5, 3, 3, 5, 7, 10, 9, 7, 4, 5, 7, 11, 12, 12, 8, 6, 7, 12, 14, 17, 15, 11, 8, 12, 15, 20, 21, 19, 13, 13, 16, 22, 26, 28, 23

Offset

0,27

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.: 1/product(1-x^(6+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006

a(n) ~ Gamma(6/7) * exp(Pi*sqrt(2*n/21)) / (2^(27/14) * 3^(3/7) * 7^(1/14) * Pi^(1/7) * n^(13/14)) * (1 - (39*sqrt(3/14)/(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} A284105(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

Example

a(45)=3 because we have 45=27+6+6+6=20+13+6+6=13+13+13+6.

Maple

g:=1/product(1-x^(6+7*j), j=0..20): gser:=series(g, x=0, 98): seq(coeff(gser, x, n), n=0..95); # Emeric Deutsch, Apr 14 2006

Mathematica

nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+6)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

Crossrefs

Cf. A284105.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), A109702 (m=6), this sequence (m=7).

Sequence in context: A136567 A336569 A324904 * A035468 A369311 A263860

Adjacent sequences: A109705 A109706 A109707 * A109709 A109710 A109711

Keyword

nonn

Author

Erich Friedman, Aug 07 2005

Extensions

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

Status

approved