A299033 a(n) = n! * [x^n] Product_{k>=1} (1 - x^k)^(n/k).

1, -1, 0, 15, -136, 885, -4896, 43085, -787200, 7775271, 326355200, -22138191801, 781498160640, -18924340012435, 239123351330304, 5915023788331125, -568462201562300416, 25327272129182225295, -795994018378027868160, 15538852668590468027711

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = n! * [x^n] exp(-n*Sum_{k>=1} d(k)*x^k/k), where d(k) is the number of divisors of k (A000005).

Example

The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 - x^k)^(n/k) begins:
n = 0: (1),  0,    0,    0,     0,      0,      0,  ...
n = 1:  1, (-1),  -1,    1,    -1,     41,   -131,  ...
n = 2:  1,  -2,   (0),   8,    -4,     72,   -704,  ...
n = 3:  1,  -3,    3,  (15),  -45,     63,  -1539,  ...
n = 4:  1,  -4,    8,   16, (-136),   224,  -1856,  ...
n = 5:  1,  -5,   15,    5,  -265,   (885), -2075,  ...
n = 6:  1,  -6,   24,  -24,  -396,   2376, (-4896), ...

Mathematica

Table[n! SeriesCoefficient[Product[(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

Crossrefs

Cf. A000005, A028343, A281267, A299034.

Sequence in context: A254409 A155648 A126536 * A030056 A225978 A175707

Adjacent sequences: A299030 A299031 A299032 * A299034 A299035 A299036

Keyword

sign

Author

Ilya Gutkovskiy, Feb 01 2018

Status

approved