A300456 - OEIS

%I #10 Aug 26 2019 06:15:38

%S 1,2,16,200,3264,65752,1565744,42878432,1324344832,45464289482,

%T 1715228012048,70471268834936,3129746696619072,149318596196238328,

%U 7612660420021177200,412865831480749700928,23725813528034949148672,1439701175150489313314864,91967625580609006328344400,6167733266497532499924699672

%N a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

%H Vaclav Kotesovec, <a href="/A300456/b300456.txt">Table of n, a(n) for n = 0..377</a>

%F a(n) ~ exp(2*sqrt(2*n) - 1) * n^(n - 3/4) / (2^(3/4)*sqrt(Pi)). - _Vaclav Kotesovec_, Aug 26 2019

%e The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(n^k) begins:

%e n = 0: (1),  0,    0,    0,     0,       0,  ...

%e n = 1:  1,  (2),   4,    8,    14,      24,  ...

%e n = 2:  1,   4,  (16),  60,   208,     692,  ...

%e n = 3:  1,   6,   36, (200), 1038,    5160   ...

%e n = 4:  1,   8,   64,  472, (3264),  21608,  ...

%e n = 5:  1,  10,  100,  920,  7950,  (65752), ...

%t Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

%Y Cf. A015128, A252654, A261519, A261520, A270919, A270923, A270924, A292805, A300457, A300458.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Mar 06 2018