OFFSET
0,2
FORMULA
a(n) ~ c * d^n / sqrt(n), where d = 7.862983395705905261519347909953827161057584... and c = 0.23317816342157644853479309078... - Vaclav Kotesovec, May 04 2018
EXAMPLE
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), ...
MATHEMATICA
Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 23}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 24 2018
STATUS
approved