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A224364 G.f.: exp( Sum_{n>=1} A064027(n)*x^n/n ), where A064027(n) = (-1)^n*Sum_{d|n}(-1)^d*d^2.

%I A224364 #25 Jan 17 2019 09:21:51
%S A224364 1,1,2,5,10,18,32,59,106,181,305,518,867,1418,2301,3724,5966,9448,
%T A224364 14862,23263,36165,55802,85609,130732,198574,299941,450946,675153,
%U A224364 1006395,1493598,2207928,3251926,4771934,6977018,10166502,14766512,21379861,30859013,44409543,63729443
%N A224364 G.f.: exp( Sum_{n>=1} A064027(n)*x^n/n ), where A064027(n) = (-1)^n*Sum_{d|n}(-1)^d*d^2.
%C A224364 "Number of the pyramid partition of pyramid partitions obtained from the fundamental pyramid partition after the removal of n bricks." [From the Bouttier et al. reference] - _Joerg Arndt_, Jul 03 2014
%D A224364 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.
%H A224364 Seiichi Manyama, <a href="/A224364/b224364.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Paul D. Hanna)
%H A224364 Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel, <a href="http://arxiv.org/abs/1407.0665">From Aztec diamonds to pyramids: steep tilings</a>, arXiv:1407.0665 [math.CO], 2014-2017.
%H A224364 Balázs Szendrői, <a href="http://dx.doi.org/10.2140/gt.2008.12.1171">Non-commutative Donaldson-Thomas invariants and the conifold</a>, Geometry & Topology 12.2 (2008): 1171-1202.
%H A224364 B. Young, <a href="https://arxiv.org/abs/0709.3079">Computing a pyramid partition generating function with dimer shuffling</a>, arXiv:0709.3079 [math.CO], 2007-2008.
%H A224364 B. Young, <a href="https://doi.org/10.1016/j.jcta.2008.06.006">Computing a pyramid partition generating function with dimer shuffling</a>, Journal of Combinatorial Theory, Series A, Volume 116, Issue 2, February 2009, Pages 334-350.
%H A224364 B. Young, J. Bryan, <a href="https://arxiv.org/abs/0802.3948">Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds</a>, arXiv:0802.3948 [math.CO], 2008.
%F A224364 G.f.: Product_{k>=1} (1+q^(2*k-1))^(2*k-1) / (1-q^(2*k))^(2*k). - _Joerg Arndt_, Jul 03 2014
%F A224364 a(n) ~ exp(1/6 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(2/9) / (2^(25/36) * A^2 * sqrt(3*Pi) * n^(13/18)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 14 2017
%e A224364 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 +...
%e A224364 where
%e A224364 log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + 50*x^7/7 + 83*x^8/8 + 91*x^9/9 + 78*x^10/10 +...+ A064027(n)*x^n/n +...
%t A224364 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1) / (1 - x^(2*k))^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 14 2017 *)
%o A224364 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, sumdiv(k, d, (-1)^d*d^2)*(-x)^k/k)+x*O(x^n)), n)}
%o A224364 for(n=0, 40, print1(a(n), ", "))
%o A224364 (PARI) N=66; q='q+O('q^N); Vec(prod(k=1,N, (1+q^(2*k-1))^(2*k-1)/(1-q^(2*k))^(2*k) )) \\ _Joerg Arndt_, Jul 03 2014
%Y A224364 Cf. A064027.
%K A224364 nonn
%O A224364 0,3
%A A224364 _Paul D. Hanna_, Apr 04 2013