A279328 Expansion of Product_{k>=1} (1 + x^(2*k)) / (1 - x^k).

1, 1, 3, 4, 8, 11, 20, 27, 44, 60, 92, 124, 183, 244, 348, 461, 640, 840, 1144, 1488, 1992, 2572, 3393, 4348, 5668, 7212, 9301, 11760, 15024, 18880, 23924, 29892, 37596, 46728, 58376, 72193, 89644, 110340, 136248, 166968, 205115, 250316, 306056, 372032, 452876

Offset: 0

Author

Seiichi Manyama, Dec 09 2016

Keywords

• nonn

Examples

			G.f.: 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 11*x^5 + 20*x^6 + 27*x^7 + 44*x^8 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from Seiichi Manyama)
• Aidan Carlson, Brian Hopkins, and James A. Sellers, Enumeration modulo four of overpartitions wherein only even parts may be overlined, Disc. Math. Lett. (2024) Vol. 14, 95-102. See p. 96.

Crossrefs

Cf. Product_{k>=1} (1 + x^(m*k)) / (1 - x^k): A015128 (m=1), this sequence (m=2), A266648 (m=3).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(2-irem(i, 2))*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 03 2025

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k)) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 10 2016 *)

Formula

G.f.: Product_{k>=1} (1 + x^(2*k)) / (1 - x^k).

a(n) ~ sqrt(5/6) * exp(sqrt(5*n/6)*Pi) / (8*n). - Vaclav Kotesovec, Dec 10 2016