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%I #26 Nov 23 2019 09:27:24
%S 1,1,4,5,14,18,41,54,109,145,267,357,618,826,1359,1815,2872,3824,5859,
%T 7774,11600,15329,22362,29425,42113,55167,77648,101267,140479,182395,
%U 249789,322906,437199,562755,754171,966713,1283630,1638716,2157763
%N Expansion of q^(5/24) / (eta(q) * eta(q^2)^2) in powers of q
%C Number of partitions of n where there are 3 kinds of even parts. - _Ilya Gutkovskiy_, Jan 17 2018
%C 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)
%H Seiichi Manyama, <a href="/A029862/b029862.txt">Table of n, a(n) for n = 0..10000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Euler transform of period 2 sequence [ 1, 3, ...].
%F G.f.: Product_{k>0} 1 / ((1 - x^(2*k))^3 * (1 - x^(2*k-1))). - _Michael Somos_, Mar 23 2003
%F a(n) ~ exp(2*Pi*sqrt(n/3))/(6*sqrt(2)*n^(3/2)). - _Vaclav Kotesovec_, Sep 07 2015
%e 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 + ...
%e G.f. = q^-5 + q^19 + 4*q^43 + 5*q^67 + 14*q^91 + 18*q^115 + 41*q^139 + ...
%e From _Gus Wiseman_, Oct 27 2018: (Start)
%e 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:
%e {{1}} {{1,1}} {{1},{2,2}} {{1,1},{2,2}} {{1},{2,2},{3,3}}
%e {{1,2}} {{1},{2,3}} {{1,2},{1,2}} {{1},{2,3},{2,3}}
%e {{1},{1}} {{2},{1,2}} {{1,2},{3,3}} {{1},{2,3},{4,4}}
%e {{1},{2}} {{1},{2},{2}} {{1,2},{3,4}} {{1},{2,3},{4,5}}
%e {{1},{2},{3}} {{1,3},{2,3}} {{1},{2,4},{3,4}}
%e {{1},{1},{2,2}} {{2},{1,2},{3,3}}
%e {{1},{1},{2,3}} {{2},{1,3},{2,3}}
%e {{1},{2},{1,2}} {{4},{1,2},{3,4}}
%e {{1},{2},{3,3}} {{1},{1},{3},{2,3}}
%e {{1},{2},{3,4}} {{1},{2},{2},{3,3}}
%e {{1},{3},{2,3}} {{1},{2},{2},{3,4}}
%e {{1},{1},{2},{2}} {{1},{2},{3},{2,3}}
%e {{1},{2},{3},{3}} {{1},{2},{3},{4,4}}
%e {{1},{2},{3},{4}} {{1},{2},{3},{4,5}}
%e {{1},{2},{4},{3,4}}
%e {{1},{2},{2},{3},{3}}
%e {{1},{2},{3},{4},{4}}
%e {{1},{2},{3},{4},{5}}
%e (End)
%t 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 *)
%t QP = QPochhammer; s = 1/(QP[q]*QP[q^2]^2) + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%o (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))};
%Y Cf. A001358, A007716, A007717, A037143, A320462, A320655, A320663, A320665.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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