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
Keywords
Examples
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), ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Programs
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Mathematica
Table[n! SeriesCoefficient[Product[(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
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