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URL: https://oeis.org/A109702

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A109702
Number of partitions of n into parts each equal to 5 mod 6.
8
1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 2, 1, 2, 3, 4, 4, 2, 2, 3, 5, 6, 5, 3, 3, 5, 7, 8, 6, 4, 5, 8, 10, 10, 8, 6, 8, 11, 13, 13, 10, 9, 12, 15, 18, 17, 14, 13, 16, 21, 23, 22, 18, 18, 23, 28, 31, 28, 24, 25, 31, 38, 39, 36, 32, 34
OFFSET
0,23
LINKS
FORMULA
G.f.: 1/product(1-x^(5+6j),j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(5/6) * exp(Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(1/6) * n^(11/12)) * (1 - (55/(24*Pi) + Pi/144) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284104(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
Euler transform of period 6 sequence [ 0, 0, 0, 0, 1, 0, ...]. - Kevin T. Acres, Apr 28 2018
EXAMPLE
a(40)=4 since 40 = 35+5 = 29+11 = 23+17 = 5+5+5+5+5+5+5+5.
MAPLE
g:=1/product(1-x^(5+6*j), j=0..20): gser:=series(g, x=0, 92): seq(coeff(gser, x, n), n=0..89); # Emeric Deutsch, Apr 14 2006
MATHEMATICA
nmax=100; CoefficientList[Series[Product[1/(1-x^(6*k+5)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)
Table[Count[IntegerPartitions[n], _?(Union[Mod[#, 6]]=={5}&)], {n, 0, 90}] (* Harvey P. Dale, Mar 08 2022 *)
CROSSREFS
Cf. A284104.
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), this sequence (m=6), A109708 (m=7).
Sequence in context: A133734 A353315 A344164 * A115412 A290217 A293285
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