A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).

1, -1, 2, -5, 18, -60, 189, -601, 1967, -6544, 21872, -73247, 246080, -829924, 2808357, -9527485, 32389671, -110316862, 376372802, -1286063899, 4400499380, -15075608840, 51704898623, -177513230200, 610007283817, -2098029341745, 7221561430933, -24875274224531

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Formula

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.50975432794970334043727352337... and c = 0.2457469629428839220188283... - Vaclav Kotesovec, Aug 21 2018

Example

a(0) = 1;
a(1) = [x^1] 1/(1 + x) = -1;
a(2) = [x^2] 1/((1 + x)^2*(1 + x^2)) = 2;
a(3) = [x^3] 1/((1 + x)^3*(1 + x^2)^2*(1 + x^3)) = -5;
a(4) = [x^4] 1/((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4)) = 18;
a(5) = [x^5] 1/((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5)) = -60, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + x^k)^(n-k+1) begins:
n = 0: (1),  0,   0,    0,   0,    0,  ...
n = 1:  1, (-1),  1,   -1,   1,   -1,  ...
n = 2:  1,  -2,  (2),  -2,   3,   -4,  ...
n = 3:  1,  -3,   4,  (-5),  9,  -14,  ...
n = 4:  1,  -4,   7,  -10, (18), -30,  ...
n = 5:  1,  -5,  11,  -18,  33, (-60), ...

Mathematica

Table[SeriesCoefficient[Product[1/(1 + x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 27}]

Crossrefs

Cf. A206228, A206229, A255526, A255528, A303173.

Sequence in context: A148424 A148425 A286317 * A148426 A321099 A241894

Adjacent sequences: A303171 A303172 A303173 * A303175 A303176 A303177

Keyword

sign

Author

Ilya Gutkovskiy, Apr 19 2018

Status

approved