A246584 - OEIS

%I #34 Oct 27 2022 12:04:58

%S 1,2,6,12,26,48,92,160,282,470,784,1260,2020,3152,4896,7456,11290,

%T 16836,24962,36556,53232,76736,110012,156384,221156,310482,433776,

%U 602200,832224,1143696,1565088,2131072,2890266,3902344,5249356,7032576,9389022,12488368

%N Number of overcubic partitions of n.

%C Convolution of A001935 and A002513. - _Vaclav Kotesovec_, Aug 16 2019

%H Alois P. Heinz, <a href="/A246584/b246584.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael D. Hirschhorn, <a href="https://www.thebookshelf.auckland.ac.nz/docs/NZJMaths/nzjmaths042/nzjmaths042-00-021.pdf">A note on overcubic partitions</a>, New Zealand J. Math., 42:229-234, 2012.

%H Bernard L. S. Lin, <a href="https://doi.org/10.37236/4400">Arithmetic properties of overcubic partition pairs</a>, Electronic Journal of Combinatorics 21(3) (2014), #P3.35.

%H James A. Sellers, <a href="https://ajc.maths.uq.edu.au/pdf/60/ajc_v60_p191.pdf">Elementary proofs of congruences for the cubic and overcubic partition functions</a>, Australasian Journal of Combinatorics, 60(2) (2014), 191-197.

%F G.f.: Product_{k>=1} (1+x^k) * (1+x^(2*k)) / ((1-x^k) * (1-x^(2*k))). - _Vaclav Kotesovec_, Aug 16 2019

%F a(n) ~ 3^(3/4) * exp(sqrt(3*n/2)*Pi) / (2^(19/4)*n^(5/4)). - _Vaclav Kotesovec_, Aug 16 2019

%p # to get 140 terms:

%p ph:=add(q^(n^2),n=-12..12);

%p ph:=series(ph,q,140);

%p g1:=1/(subs(q=-q,ph)*subs(q=-q^2,ph));

%p g1:=series(g1,q,140);

%p seriestolist(%);

%p # second Maple program:

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*

%p       `if`(irem(d, 4)=2, 3, 2), d=divisors(j)), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Aug 17 2019

%t 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 *)

%t 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 *)

%Y Trisections: A246585, A246586, A246587.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Sep 03 2014