A029862 Expansion of q^(5/24) / (eta(q) * eta(q^2)^2) in powers of q.

1, 1, 4, 5, 14, 18, 41, 54, 109, 145, 267, 357, 618, 826, 1359, 1815, 2872, 3824, 5859, 7774, 11600, 15329, 22362, 29425, 42113, 55167, 77648, 101267, 140479, 182395, 249789, 322906, 437199, 562755, 754171, 966713, 1283630, 1638716, 2157763

Offset

0,3

Comments

Number of partitions of n where there are 3 kinds of even parts. - Ilya Gutkovskiy, Jan 17 2018

Also the number of non-isomorphic multiset partitions of weight n using singletons or pairs where no vertex appears more than twice. - Gus Wiseman, Oct 18 2018 (Proved by Andrew Howroyd, Oct 26 2018)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Transforms

Formula

Euler transform of period 2 sequence [ 1, 3, ...].

G.f.: Product_{k>0} 1 / ((1 - x^(2*k))^3 * (1 - x^(2*k-1))). - Michael Somos, Mar 23 2003

a(n) ~ exp(2*Pi*sqrt(n/3))/(6*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Sep 07 2015

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 2 * exp(-5*Pi/24) * 2^(3/8) * Gamma(3/4)^3 / Pi^(3/4) = A388385. - Simon Plouffe, Sep 15 2025

Example

G.f. = 1 + x + 4*x^2 + 5*x^3 + 14*x^4 + 18*x^5 + 41*x^6 + 54*x^7 + 109*x^8 + ...
G.f. = q^-5 + q^19 + 4*q^43 + 5*q^67 + 14*q^91 + 18*q^115 + 41*q^139 + ...
From Gus Wiseman, Oct 27 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 multiset partitions using singletons or pairs where no vertex appears more than twice:
  {{1}}  {{1,1}}    {{1},{2,2}}    {{1,1},{2,2}}      {{1},{2,2},{3,3}}
         {{1,2}}    {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3},{2,3}}
         {{1},{1}}  {{2},{1,2}}    {{1,2},{3,3}}      {{1},{2,3},{4,4}}
         {{1},{2}}  {{1},{2},{2}}  {{1,2},{3,4}}      {{1},{2,3},{4,5}}
                    {{1},{2},{3}}  {{1,3},{2,3}}      {{1},{2,4},{3,4}}
                                   {{1},{1},{2,2}}    {{2},{1,2},{3,3}}
                                   {{1},{1},{2,3}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{4},{1,2},{3,4}}
                                   {{1},{2},{3,3}}    {{1},{1},{3},{2,3}}
                                   {{1},{2},{3,4}}    {{1},{2},{2},{3,3}}
                                   {{1},{3},{2,3}}    {{1},{2},{2},{3,4}}
                                   {{1},{1},{2},{2}}  {{1},{2},{3},{2,3}}
                                   {{1},{2},{3},{3}}  {{1},{2},{3},{4,4}}
                                   {{1},{2},{3},{4}}  {{1},{2},{3},{4,5}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{2},{3},{3}}
                                                      {{1},{2},{3},{4},{4}}
                                                      {{1},{2},{3},{4},{5}}
(End)

Mathematica

nmax = 40; CoefficientList[Series[Product[1 / ((1 - x^(2*k))^3 * (1 - x^(2*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
QP = QPochhammer; s = 1/(QP[q]*QP[q^2]^2) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (eta(x + A) * eta(x^2 + A)^2), n))};

Crossrefs

Cf. A001358, A007716, A007717, A037143, A320462, A320655, A320663, A320665.

Sequence in context: A222364 A222372 A302348 * A092432 A041669 A300278

Adjacent sequences: A029859 A029860 A029861 * A029863 A029864 A029865

Keyword

nonn

Author

N. J. A. Sloane

Status

approved