A266648 Expansion of Product_{k>=1} (1 + x^(3*k)) / (1 - x^k).
1, 1, 2, 4, 6, 9, 15, 21, 31, 46, 64, 89, 126, 170, 231, 314, 417, 552, 733, 955, 1244, 1617, 2079, 2665, 3413, 4331, 5485, 6931, 8704, 10901, 13629, 16949, 21033, 26045, 32123, 39529, 48553, 59429, 72599, 88518, 107624, 130599, 158209, 191175, 230611, 277717, 333730, 400375, 479598, 573386, 684481
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
- Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020, p. 6.
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(irem(i, 3)=0, 2, 1)*add(b(n-i*j, i-1), j=1..n/i))) end: a:= n-> b(n$2): seq(a(n), n=0..50); # Alois P. Heinz, Feb 03 2025 -
Mathematica
nmax = 50; CoefficientList[Series[Product[(1+x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Formula
a(n) ~ sqrt(7) * exp(sqrt(7*n)*Pi/3) / (24*n).
Comments