A299034 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k)^(n/k).

1, 1, 8, 93, 1544, 32615, 843264, 25739539, 906373376, 36163950849, 1612483625600, 79458277381901, 4288069172500992, 251520785449249927, 15932801526165085184, 1084003570689331039875, 78835487923639854792704, 6103175938145968656408641, 501114006272655771562911744

Offset

0,3

Links

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

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).

a(n) ~ c * d^n * n^n, where d = 1.7257974131308983723949107467... and c = 0.693704376971941705824592525... - Vaclav Kotesovec, Sep 08 2018

Example

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),  3,   11,    59,    339,    2629,  ...
n = 2:  1,  2,  (8),  40,   260,   1928,   17056,  ...
n = 3:  1,  3,  15,  (93),  711,   6237,   62901,  ...
n = 4:  1,  4,  24,  176, (1544), 15456,  174784,  ...
n = 5:  1,  5,  35,  295,  2915, (32615), 407725,  ...
n = 6:  1,  6,  48,  456,  5004,  61704, (843264), ...

Mathematica

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

Crossrefs

Cf. A000005, A028342, A255672, A299033.

Sequence in context: A194043 A122419 A319176 * A299339 A299846 A214385

Adjacent sequences: A299031 A299032 A299033 * A299035 A299036 A299037

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 01 2018

Status

approved