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

Author

Jeremy Lovejoy, Aug 06 2015

Keywords

• nonn

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.

Programs

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 *)

Formula

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

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