A300187 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k)^(n/k).

1, 1, 4, 39, 488, 7615, 147024, 3371137, 89079808, 2665537713, 89142430400, 3295096700071, 133399600068096, 5870116973678191, 278971698167158528, 14239859507270510625, 776985219329347518464, 45130494178637796970273, 2780224621391401396134912, 181059775626543107582734183

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} A048272(k)*x^k/k).

a(n) ~ c * d^n * n^n, where d = 1.294982800733109500251... and c = 0.6755467963500480915... - Vaclav Kotesovec, Sep 07 2018

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,    5,    11,     59,     439,  ...
n = 2:  1,  2,  (4),  16,    68,    328,    2416,  ...
n = 3:  1,  3,   9,  (39),  207,   1197,    8811,  ...
n = 4:  1,  4,  16,   80,  (488),  3296,   25984,  ...
n = 5:  1,  5,  25,  145,   995,  (7615),  65575,  ...
n = 6:  1,  6,  36,  240,  1836,  15624, (147024), ...

Mathematica

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

Crossrefs

Cf. A048272, A168243, A270922, A299033, A299034, A300188.

Sequence in context: A203211 A214823 A275517 * A068187 A307698 A319177

Adjacent sequences: A300184 A300185 A300186 * A300188 A300189 A300190

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 28 2018

Status

approved