A260890 The number of overpartitions of n with restricted odd differences.

1, 1, 3, 3, 8, 9, 18, 21, 39, 46, 78, 93, 150, 180, 276, 333, 494, 597, 858, 1038, 1458, 1764, 2424, 2931, 3960, 4783, 6360, 7671, 10068, 12123, 15720, 18894, 24249, 29088, 36978, 44268, 55808, 66672, 83406, 99435, 123540, 146973, 181440, 215406, 264390, 313236, 382404, 452130, 549258

Offset

0,3

Comments

The number of overpartitions of n where (i) the difference between successive parts may be odd only if the larger is overlined and (ii) if the smallest part is overlined, then it is odd.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, Overpartitions with restricted odd differences, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

Formula

G.f.: Product_{n >= 1} (1-q^(3n))/((1-q^n)*(1-q^(2*n)).

a(n) ~ sqrt(21) * exp(Pi*sqrt(7*n)/3) / (36*n). - Vaclav Kotesovec, Jun 13 2019

Maple

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*[1, 1, 2, 0, 2, 1]
      [irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 08 2015

Mathematica

QP = QPochhammer; QP[q^3]/(QP[q] QP[q^2]) + O[q]^50 // CoefficientList[#, q]& (* Jean-François Alcover, Mar 23 2017 *)

Crossrefs

Sequence in context: A363125 A335602 A092549 * A022663 A304967 A323654

Adjacent sequences: A260887 A260888 A260889 * A260891 A260892 A260893

Keyword

nonn

Author

Jeremy Lovejoy, Aug 06 2015

Status

approved