A109707 Number of partitions of n into parts each equal to 5 mod 7.
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, 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, 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
Offset: 0
Keywords
Examples
a(36)=3 because we have 36=26+5+5=19+12+5=12+12+12.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
- M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Programs
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Maple
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
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Mathematica
nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+5)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)
Formula
G.f.: 1/product(1-x^(5+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006
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
a(n) = (1/n)*Sum_{k=1..n} A284446(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 28 2017
Extensions
Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015