cp's OEIS Frontend

A147785 Number of partitions of n into parts divisible by 3 or 5.

%I A147785 #17 Dec 01 2018 10:14:39
%S A147785 1,0,0,1,0,1,2,0,1,3,2,2,5,2,3,9,4,5,13,6,11,19,10,15,28,19,23,40,27,
%T A147785 34,63,40,50,85,59,79,121,85,109,166,132,155,230,180,216,325,255,300,
%U A147785 436,351,429,588,485,576,789,680,784,1050,912,1053,1421,1228
%N A147785 Number of partitions of n into parts divisible by 3 or 5.
%C A147785 Also number of partitions of n with no part and no difference between two parts equal to 1,2,4 or 7.
%C A147785 Also number of partitions of n with no part appearing 1,2,4 or 7 times.
%H A147785 Alois P. Heinz, <a href="/A147785/b147785.txt">Table of n, a(n) for n = 0..10000</a>
%H A147785 A. E. Holroyd, <a href="http://arxiv.org/abs/0706.2282">Partition Identities and the Coin Exchange Problem</a>, arXiv:0706.2282 [math.CO], 2007.
%H A147785 A. E. Holroyd, <a href="https://doi.org/10.1016/j.jcta.2007.12.003">Partition Identities and the Coin Exchange Problem</a>, J. Combin. Theory Ser. A, 115 (2008) 1096-1101.
%H A147785 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 17.
%F A147785 G.f.: Product_{k>=1} (1-x^(15k)) / ((1-x^(3k))*(1-x^(5k))).
%F A147785 a(n) ~ sqrt(7/5) * exp(sqrt(14*n/5)*Pi/3) / (12*n). - _Vaclav Kotesovec_, Sep 23 2015
%p A147785 with(numtheory):
%p A147785 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A147785       `if`(irem(d, 3)=0 or irem(d, 5)=0, d, 0),
%p A147785            d=divisors(j))*a(n-j), j=1..n)/n)
%p A147785     end:
%p A147785 seq(a(n), n=0..65);  # _Alois P. Heinz_, Dec 02 2016
%t A147785 nmax = 60; CoefficientList[Series[Product[(1-x^(15*k))/((1-x^(3*k))*(1-x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 23 2015 *)
%Y A147785 Cf. A007690, A147783, A147784, A147786, A147787.
%K A147785 nonn
%O A147785 0,7
%A A147785 Alexander E. Holroyd (holroyd at math.ubc.ca)