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

0,3

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.

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = exp(Pi / 24) * Gamma(3/4) / Pi^(1/4) = A388962. - Simon Plouffe, Sep 21 2025

Example

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 + ...

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 *)

Crossrefs

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

Sequence in context: A299069 A097497 A332681 * A006167 A137504 A173401

Adjacent sequences: A279325 A279326 A279327 * A279329 A279330 A279331

Keyword

nonn

Author

Seiichi Manyama, Dec 09 2016

Status

approved