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