A224364 G.f.: exp( Sum_{n>=1} A064027(n)*x^n/n ), where A064027(n) = (-1)^n*Sum_{d|n}(-1)^d*d^2.
1, 1, 2, 5, 10, 18, 32, 59, 106, 181, 305, 518, 867, 1418, 2301, 3724, 5966, 9448, 14862, 23263, 36165, 55802, 85609, 130732, 198574, 299941, 450946, 675153, 1006395, 1493598, 2207928, 3251926, 4771934, 6977018, 10166502, 14766512, 21379861, 30859013, 44409543, 63729443
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 +... where 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 +...
References
- Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)
- Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel, From Aztec diamonds to pyramids: steep tilings, arXiv:1407.0665 [math.CO], 2014-2017.
- Balázs Szendrői, Non-commutative Donaldson-Thomas invariants and the conifold, Geometry & Topology 12.2 (2008): 1171-1202.
- B. Young, Computing a pyramid partition generating function with dimer shuffling, arXiv:0709.3079 [math.CO], 2007-2008.
- B. Young, Computing a pyramid partition generating function with dimer shuffling, Journal of Combinatorial Theory, Series A, Volume 116, Issue 2, February 2009, Pages 334-350.
- B. Young, J. Bryan, Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds, arXiv:0802.3948 [math.CO], 2008.
Crossrefs
Cf. A064027.
Programs
-
Mathematica
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 *) -
PARI
{a(n)=polcoeff(exp(sum(k=1, n, sumdiv(k, d, (-1)^d*d^2)*(-x)^k/k)+x*O(x^n)), n)} for(n=0, 40, print1(a(n), ", ")) -
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
Formula
G.f.: Product_{k>=1} (1+q^(2*k-1))^(2*k-1) / (1-q^(2*k))^(2*k). - Joerg Arndt, Jul 03 2014
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
Comments