A278769 - OEIS

%I #11 May 08 2017 00:17:45

%S 1,1,8,26,88,269,843,2456,7115,19892,54756,147355,390517,1017091,

%T 2612670,6617641,16556913,40933339,100104289,242276236,580718077,

%U 1379161494,3247074738,7581837910,17564867853,40388447308,92206496318,209069338580,470944571003,1054178579266,2345477963043,5188246121144,11412352653001

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(5*k-3)/2).

%C Euler transform of the heptagonal numbers (A000566).

%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/HeptagonalNumber.html">Heptagonal 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*(5*k-3)/2).

%F a(n) ~ exp(-3*Zeta'(-1)/2 - 5*Zeta(3)/(8*Pi^2) - 81*Zeta(3)^3/(2*Pi^8) - 3^(13/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - 3^(3/2)*Zeta(3)/(sqrt(2)*Pi^2) * sqrt(n) + 2^(7/4)*Pi/3^(5/4) * n^(3/4)) / (2^(51/32) * 3^(3/32) * Pi^(1/8) * n^(19/32)). - _Vaclav Kotesovec_, Dec 02 2016

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       d^2*(5*d-3)/2, 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 (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000294, A000566, A000335, A023871.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 28 2016