OFFSET
0,2
COMMENTS
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael D. Hirschhorn, A note on overcubic partitions, New Zealand J. Math., 42:229-234, 2012.
Bernard L. S. Lin, Arithmetic properties of overcubic partition pairs, Electronic Journal of Combinatorics 21(3) (2014), #P3.35.
James A. Sellers, Elementary proofs of congruences for the cubic and overcubic partition functions, Australasian Journal of Combinatorics, 60(2) (2014), 191-197.
FORMULA
G.f.: Product_{k>=1} (1+x^k) * (1+x^(2*k)) / ((1-x^k) * (1-x^(2*k))). - Vaclav Kotesovec, Aug 16 2019
a(n) ~ 3^(3/4) * exp(sqrt(3*n/2)*Pi) / (2^(19/4)*n^(5/4)). - Vaclav Kotesovec, Aug 16 2019
MAPLE
# to get 140 terms:
ph:=add(q^(n^2), n=-12..12);
ph:=series(ph, q, 140);
g1:=1/(subs(q=-q, ph)*subs(q=-q^2, ph));
g1:=series(g1, q, 140);
seriestolist(%);
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
`if`(irem(d, 4)=2, 3, 2), d=divisors(j)), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Aug 17 2019
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k)) / ((1-x^k) * (1-x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 16 2019 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(2*k)) / (1-x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 16 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 03 2014
STATUS
approved