A300188 a(n) = n! * [x^n] Product_{k>=1} 1/(1 + x^k)^(n/k).

1, -1, 4, -39, 536, -9115, 185904, -4461877, 123647488, -3886461081, 136538590400, -5300491027711, 225313697972736, -10409021924850211, 519298241645107456, -27824560148201248125, 1593597443825288904704, -97153909607626767338353, 6281720886474120790582272

Offset: 0

Author

Ilya Gutkovskiy, Feb 28 2018

Keywords

• sign

Examples

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 + x^k)^(n/k) begins:
n = 0: (1),  0,   0,     0,     0,       0,        0,  ...
n = 1:  1, (-1),  1,    -5,    23,    -119,      619,  ...
n = 2:  1,  -2,  (4),  -16,    92,    -568,     3856,  ...
n = 3:  1,  -3,   9,  (-39),  243,   -1737,    13671,  ...
n = 4:  1,  -4,  16,   -80,  (536),  -4256,    37504,  ...
n = 5:  1,  -5,  25,  -145,  1055,  (-9115),   88075,  ...
n = 6:  1,  -6,  36,  -240,  1908,  -17784,  (185904), ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A048272, A281266, A294356, A299033, A299034, A300187.

Programs

Mathematica

Table[n! SeriesCoefficient[Product[1/(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] exp(-n*Sum_{k>=1} A048272(k)*x^k/k).

a(n) ~ (-1)^n * c * d^n * n^n, where d = 1.3587950730244927060955... and c = 0.6449711831436950784... - Vaclav Kotesovec, Sep 08 2018