A246584 Number of overcubic partitions of n.

1, 2, 6, 12, 26, 48, 92, 160, 282, 470, 784, 1260, 2020, 3152, 4896, 7456, 11290, 16836, 24962, 36556, 53232, 76736, 110012, 156384, 221156, 310482, 433776, 602200, 832224, 1143696, 1565088, 2131072, 2890266, 3902344, 5249356, 7032576, 9389022, 12488368

Offset

0,2

Comments

Convolution of A001935 and A002513. - Vaclav Kotesovec, Aug 16 2019

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

Trisections: A246585, A246586, A246587.

Sequence in context: A141347 A335724 A300120 * A054454 A084170 A245264

Adjacent sequences: A246581 A246582 A246583 * A246585 A246586 A246587

Keyword

nonn

Author

N. J. A. Sloane, Sep 03 2014

Status

approved