A308337
Expansion of e.g.f. exp(-1 + Product_{k>=1} 1/(1 - x^k/k)).
Original entry on oeis.org
1, 1, 4, 21, 146, 1210, 11894, 133000, 1677668, 23439684, 359651232, 6000427884, 108157599120, 2092493844480, 43240336406136, 950021322002904, 22108068716440272, 543048769028068944, 14037848562783308160, 380851191359082217488, 10818262594065294568416
Offset: 0
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nmax = 20; CoefficientList[Series[Exp[Product[1/(1 - x^k/k), {k, 1, nmax}] - 1], {x, 0, nmax}], x] Range[0, nmax]!
A308336
Expansion of e.g.f. exp(-1 + Product_{k>=1} (1 + x^k/k!)).
Original entry on oeis.org
1, 1, 2, 8, 31, 147, 884, 5567, 39176, 311400, 2644490, 24206327, 239684768, 2519262527, 28077597357, 331892965533, 4130002336563, 53944450834303, 738940309779760, 10577568411051305, 157846971489443335, 2452481386778640564, 39589449956634478543
Offset: 0
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nmax = 22; CoefficientList[Series[Exp[Product[(1 + x^k/k!), {k, 1, nmax}] - 1], {x, 0, nmax}], x] Range[0, nmax]!
A328193
Expansion of e.g.f. Sum_{k>=1} log(1/(1 + (-x)^k/k)).
Original entry on oeis.org
1, 0, 4, 3, 48, 10, 1440, 1890, 85120, 49896, 7257600, 6883800, 958003200, 792277200, 178919989248, 194107914000, 41845579776000, 29714949264000, 12804747411456000, 12900082757417856, 4918792391884800000, 4594737608304480000, 2248001455555215360000
Offset: 1
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a:= n-> n!*add((-1)^(n-d)/(d*(n/d)^d), d=numtheory[divisors](n)):
seq(a(n), n=1..24); # Alois P. Heinz, Oct 30 2019
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nmax = 23; CoefficientList[Series[Sum[Log[1/(1 + (-x)^k/k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(n - d)/(d (n/d)^d), {d, Divisors[n]}], {n, 1, 23}]
Showing 1-3 of 3 results.