A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).
1, 1, 9, 30, 106, 339, 1106, 3355, 10102, 29358, 83908, 234394, 644286, 1739933, 4631675, 12153197, 31485413, 80576160, 203902261, 510490213, 1265353568, 3106771717, 7559844833, 18239351931, 43650061720, 103657177941, 244346681972, 571930478187, 1329655624297, 3071230379625, 7049750442386, 16085170634548, 36489192684910
Offset: 0
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Octagonal Number
- Index to sequences related to polygonal numbers
Programs
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Maple
with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add( d^2*(3*d-2), d=divisors(j))*a(n-j), j=1..n)/n) end: seq(a(n), n=0..35); # Alois P. Heinz, Dec 02 2016 -
Mathematica
nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x] -
Python
from sympy import divisors from sympy.core.cache import cacheit @cacheit def a(n): return 1 if n==0 else sum(sum(d**2*(3*d - 2) for d in divisors(j))*a(n - j) for j in range(1, n + 1))//n print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 06 2017, after Maple code
Formula
G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).
a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) - 2*Zeta(3) * sqrt(5*n) / Pi^2 - 10*Zeta(3)^2 * (5*n)^(1/4) / Pi^5 - 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) - 1/6) * A^2 / (2^(3/2) * 5^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Comments