|
%I #11 May 08 2017 00:17:09
%S 1,1,7,22,71,206,616,1712,4743,12677,33407,86085,218677,546060,
%T 1345840,3271893,7861239,18670881,43883904,102112483,235401947,
%U 537869136,1218743007,2739566083,6111766043,13536683750,29775945929,65065819486,141285315728,304935221675,654318376244,1396166024244,2963068779402
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).
%C Euler transform of the hexagonal numbers (A000384).
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).
%F a(n) ~ exp(-Zeta'(-1) - Zeta(3)/(2*Pi^2) - 75*Zeta(3)^3/(4*Pi^8) - 15^(5/4)*Zeta(3)^2/(2^(9/4)*Pi^5) * n^(1/4) - sqrt(15/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(9/4)*Pi/(3^(5/4)*5^(1/4)) * n^(3/4)) / (2^(67/48) * 15^(5/48) * Pi^(1/12) * n^(29/48)). - _Vaclav Kotesovec_, Dec 02 2016
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p d^2*(2*d-1), d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Dec 02 2016
%t nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000294, A000384, A000335, A023871.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 28 2016
|
