A300457 -id:A300457 - cp's OEIS frontend

A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

1, 2, 16, 200, 3264, 65752, 1565744, 42878432, 1324344832, 45464289482, 1715228012048, 70471268834936, 3129746696619072, 149318596196238328, 7612660420021177200, 412865831480749700928, 23725813528034949148672, 1439701175150489313314864, 91967625580609006328344400, 6167733266497532499924699672

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

nonn

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(n^k) begins:
n = 0: (1),  0,    0,    0,     0,       0,  ...
n = 1:  1,  (2),   4,    8,    14,      24,  ...
n = 2:  1,   4,  (16),  60,   208,     692,  ...
n = 3:  1,   6,   36, (200), 1038,    5160   ...
n = 4:  1,   8,   64,  472, (3264),  21608,  ...
n = 5:  1,  10,  100,  920,  7950,  (65752), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..377

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

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

a(n) ~ exp(2*sqrt(2*n) - 1) * n^(n - 3/4) / (2^(3/4)*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019

A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

Original entry on oeis.org

1, -1, -1, -10, 11, 374, 9792, 183847, 3469427, 65038049, 1195396233, 19667738452, 189089161562, -6219720781782, -606316892131934, -35104997710496175, -1795953382595105853, -88223902016631657740, -4283800987347611165184, -207864171877269042498096, -10102590396625592962089500

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			The table of coefficients of x^k in expansion of Product_{k>=1} 1/(1 + x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),   0,   -1,   1,    -1,  ...
n = 2:  1,  -2,  (-1),  -4,   3,    -2,  ...
n = 3:  1,  -3,   -3, (-10),  6,    15,  ...
n = 4:  1,  -4,   -6,  -20, (11),  104,  ...
n = 5:  1,  -5,  -10,  -35,  20,  (374), ...

Cf. A081362, A252654, A255526, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300457.

Table[SeriesCoefficient[Product[1/(1 + x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

cp's OEIS Frontend

A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

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