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
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), 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), ...
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} 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