A300412 a(n) = [x^n] Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k.

1, 2, 16, 144, 1376, 15800, 210816, 3333372, 61688448, 1318588146, 32004369200, 869282342632, 26099925704928, 857736429098848, 30605729417479104, 1177841009504482200, 48614265201514729984, 2141639401723095243324, 100282931820560447963568, 4973060138191518242569120

Offset: 0

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

• nonn

Examples

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k begins:
n = 0: (1),  0,   0,    0,     0,       0,  ...
n = 1:  1,  (2),  6,   16,    38,      88,  ...
n = 2:  1,   4, (16),  60,   192,     596,  ...
n = 3:  1,   6,  30, (144),  582,    2280,  ...
n = 4:  1,   8,  48,  280, (1376),   6568,  ...
n = 5:  1,  10,  70,  480,  2790,  (15800), ...
		

Crossrefs

Cf. A156616, A261563, A266942, A270919, A270924, A292419, A298985, A298987.

Programs

Mathematica

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

Formula

a(n) ~ 2 * n^n * (1 + 4/n + 14/n^2 + 44/n^3 + 124/n^4 + 328/n^5 + 824/n^6 + 1980/n^7 + 4590/n^8 + 10320/n^9 + 22584/n^10 + ...), for coefficients see A261451. - Vaclav Kotesovec, Mar 05 2018