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
Keywords
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.
Programs
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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 *)
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
Comments