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A109699 Number of partitions of n into parts each equal to 3 mod 5.

%I A109699 #23 Dec 01 2024 13:01:33
%S A109699 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,
%T A109699 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,
%U A109699 35,38,36,44,42,47,51,50,58,57,63,68,66,79,76,82,92,88,101,104,107,120
%N A109699 Number of partitions of n into parts each equal to 3 mod 5.
%H A109699 Vaclav Kotesovec, <a href="/A109699/b109699.txt">Table of n, a(n) for n = 0..10000</a>
%F A109699 G.f.: 1/product(1-x^(3+5j), j=0..infinity). - _Emeric Deutsch_, Mar 30 2006
%F A109699 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
%F A109699 a(n) = (1/n)*Sum_{k=1..n} A284281(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 24 2017
%e A109699 a(21)=3 since 21 = 18+3 = 13+8 = 3+3+3+3+3+3+3
%p A109699 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
%t A109699 nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+3)),{k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 27 2015 *)
%t A109699 Join[{1},Table[Length[Select[IntegerPartitions[n],Union[Mod[#,5]]=={3}&]],{n,80}]] (* _Harvey P. Dale_, Dec 01 2024 *)
%o A109699 (PARI) Vec(prod(k=0, 100, 1/(1 - x^(5*k + 3))) + O(x^111)) \\ _Indranil Ghosh_, Mar 24 2017
%Y A109699 Cf. A284281.
%K A109699 nonn
%O A109699 0,17
%A A109699 _Erich Friedman_, Aug 07 2005
%E A109699 More terms from _Emeric Deutsch_, Mar 30 2006