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A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1). 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 14 10:58 EDT 2022. Contains 356116 sequences. (Running on oeis4.)