A109707 - OEIS

%I #20 Mar 28 2017 14:48:32

%S 1,0,0,0,0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,2,1,1,1,0,2,1,2,1,1,

%T 2,1,3,1,3,2,2,3,1,4,2,4,3,2,5,2,6,3,5,5,3,7,3,8,5,6,8,4,10,5,10,8,8,

%U 11,6,13,8,13,12,10,15,9,18,12,17,16,14,21,13,23,17,22,23,18,28,18,31,24,28

%N Number of partitions of n into parts each equal to 5 mod 7.

%H Vaclav Kotesovec, <a href="/A109707/b109707.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

%F G.f.: 1/product(1-x^(5+7j), j=0..infinity). - _Emeric Deutsch_, Apr 14 2006

%F a(n) ~ Gamma(5/7) * exp(Pi*sqrt(2*n/21)) / (2^(13/7) * 3^(5/14) * 7^(1/7) * Pi^(2/7) * n^(6/7)) * (1 + (11*Pi/(168*sqrt(42)) - 15*sqrt(6/7)/(7*Pi)) / sqrt(n)). - _Vaclav Kotesovec_, Feb 27 2015, extended Jan 24 2017

%F a(n) = (1/n)*Sum_{k=1..n} A284446(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 28 2017

%e a(36)=3 because we have 36=26+5+5=19+12+5=12+12+12.

%p g:=1/product(1-x^(5+7*j),j=0..20): gser:=series(g,x=0,95): seq(coeff(gser,x,n),n=0..92); # _Emeric Deutsch_, Apr 14 2006

%t nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+5)),{k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 27 2015 *)

%K nonn

%O 0,25

%A _Erich Friedman_, Aug 07 2005

%E Changed offset to 0 and added a(0)=1 by _Vaclav Kotesovec_, Feb 27 2015