A109699 Number of partitions of n into parts each equal to 3 mod 5.
1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 4, 2, 4, 4, 3, 5, 4, 5, 6, 5, 7, 6, 8, 8, 7, 11, 9, 10, 13, 10, 14, 14, 14, 17, 16, 19, 19, 20, 24, 21, 27, 27, 27, 33, 30, 35, 38, 36, 44, 42, 47, 51, 50, 58, 57, 63, 68, 66, 79, 76, 82, 92, 88, 101, 104, 107, 120
Offset: 0
Keywords
Examples
a(21)=3 since 21 = 18+3 = 13+8 = 3+3+3+3+3+3+3
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Crossrefs
Cf. A284281.
Programs
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Maple
g:=1/product(1-x^(3+5*j),j=0..25): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..80); # Emeric Deutsch, Mar 30 2006
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Mathematica
nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+3)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *) Join[{1},Table[Length[Select[IntegerPartitions[n],Union[Mod[#,5]]=={3}&]],{n,80}]] (* Harvey P. Dale, Dec 01 2024 *)
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PARI
Vec(prod(k=0, 100, 1/(1 - x^(5*k + 3))) + O(x^111)) \\ Indranil Ghosh, Mar 24 2017
Formula
G.f.: 1/product(1-x^(3+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006
a(n) ~ Gamma(3/5) * exp(Pi*sqrt(2*n/15)) / (2^(9/5) * 3^(3/10) * 5^(1/5) * Pi^(2/5) * n^(4/5)) * (1 + (11*Pi/(120*sqrt(30)) - 6*sqrt(6/5)/(5*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284281(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 24 2017
Extensions
More terms from Emeric Deutsch, Mar 30 2006