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

Author

N. J. A. Sloane

Keywords

• nonn

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)

Examples

			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)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• N. J. A. Sloane, Transforms

Crossrefs

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

Programs

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 *)

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))};

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