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A147785
Number of partitions of n into parts divisible by 3 or 5.
5
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, 34, 63, 40, 50, 85, 59, 79, 121, 85, 109, 166, 132, 155, 230, 180, 216, 325, 255, 300, 436, 351, 429, 588, 485, 576, 789, 680, 784, 1050, 912, 1053, 1421, 1228
OFFSET
0,7
COMMENTS
Also number of partitions of n with no part and no difference between two parts equal to 1,2,4 or 7.
Also number of partitions of n with no part appearing 1,2,4 or 7 times.
LINKS
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, arXiv:0706.2282 [math.CO], 2007.
A. E. Holroyd, Partition Identities and the Coin Exchange Problem, J. Combin. Theory Ser. A, 115 (2008) 1096-1101.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 17.
FORMULA
G.f.: Product_{k>=1} (1-x^(15k)) / ((1-x^(3k))*(1-x^(5k))).
a(n) ~ sqrt(7/5) * exp(sqrt(14*n/5)*Pi/3) / (12*n). - Vaclav Kotesovec, Sep 23 2015
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
`if`(irem(d, 3)=0 or irem(d, 5)=0, d, 0),
d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..65); # Alois P. Heinz, Dec 02 2016
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander E. Holroyd (holroyd at math.ubc.ca)
STATUS
approved