A278768 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)/2).
1, 1, 6, 18, 55, 150, 424, 1113, 2923, 7401, 18510, 45271, 109297, 259447, 608428, 1407958, 3222132, 7292198, 16340830, 36265672, 79775931, 173999194, 376497975, 808471181, 1723592762, 3649271887, 7675809680, 16043777217, 33332888108, 68853608216, 141438908854, 288994878713, 587458691042
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
- 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 arXiv version]
- 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, Pentagonal 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-1)/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 - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
Formula
G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)/2).
a(n) ~ exp(-Zeta'(-1)/2 - 3*Zeta(3)/(8*Pi^2) - 25*Zeta(3)^3/(6*Pi^8) - 5^(5/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - sqrt(5/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(7/4)*Pi/(3*5^(1/4)) * n^(3/4)) / (2^(155/96) * 5^(11/96) * Pi^(1/24) * n^(59/96)). - Vaclav Kotesovec, Dec 02 2016
Comments