A303483 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1).

1, 2, 10, 64, 436, 3072, 22096, 161148, 1187118, 8812050, 65806720, 493827256, 3720698056, 28128081912, 213258301824, 1620878656280, 12346263051028, 94221026620572, 720267101230410, 5514346833878672, 42274910234115352, 324490877248800232, 2493471670778297856, 19179885230907692452

Offset: 0

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

• nonn

Examples

			a(0) = 1;
a(1) = [x^1] (1 + x)/(1 - x) = 2;
a(2) = [x^2] ((1 + x)^2*(1 + x^2))/((1 - x)^2*(1 - x^2)) = 10;
a(3) = [x^3] ((1 + x)^3*(1 + x^2)^2*(1 + x^3))/((1 - x)^3*(1 - x^2)^2*(1 - x^3)) = 64;
a(4) = [x^4] ((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4))/((1 - x)^4*(1 - x^2)^3*(1 - x^3)^2*(1 - x^4)) = 436;
a(5) = [x^5] ((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5))/((1 - x)^5*(1 - x^2)^4*(1 - x^3)^3*(1 - x^4)^2*(1 - x^5)) = 3072, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1) begins:
n = 0: (1),  0,   0,    0,    0,     0,  ...
n = 1:  1,  (2),  2,    2,    2,     2,  ...
n = 2:  1,   4, (10),  20,   34,    52,  ...
n = 3:  1,   6,  22,  (64), 158,   346,  ...
n = 4:  1,   8,  38,  140, (436), 1200,  ...
n = 5:  1,  10,  58,  256,  946, (3072), ...
		

Crossrefs

Cf. A206228, A206229, A270919, A303173, A303174.

Programs

Mathematica

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = 7.862983395705905261519347909953827161057584... and c = 0.23317816342157644853479309078... - Vaclav Kotesovec, May 04 2018