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
Keywords
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
Programs
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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