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}]
A336306
a(n) = (n!)^n * [x^n] Product_{n>=1} (1 + x^k/k^n).
Original entry on oeis.org
1, 1, 1, 35, 5392, 35462624, 15419509448256, 445352317449860352384, 1733058447330128629281872412672, 1124641798042952855847954946807366969982976, 155064212713307814902013200520441969883490549760000000000
Offset: 0
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b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$3):
seq(a(n), n=0..12); # Alois P. Heinz, Jul 27 2023
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Table[(n!)^n SeriesCoefficient[Product[(1 + x^k/k^n), {k, 1, n}], {x, 0, n}], {n, 0, 10}]
A338439
a(n) = n! * [x^n] Product_{k=1..n, gcd(n,k) = 1} (1 + x^k/k).
Original entry on oeis.org
1, 1, 0, 3, 8, 50, 144, 2394, 8448, 89424, 576000, 20124720, 57231360, 3213905760, 11285084160, 217204092000, 2843121254400, 187660890063360, 558255985459200, 64849189355274240, 239933887119360000, 8405611881201561600, 116110668405473280000, 13912098832249673932800
Offset: 0
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Table[n! SeriesCoefficient[Product[(1 + Boole[GCD[n, k] == 1] x^k/k), {k, 1, n}], {x, 0, n}], {n, 0, 23}]
A371389
Expansion of e.g.f. Product_{k>=1} (1 + x^k/k)^2.
Original entry on oeis.org
1, 2, 4, 16, 74, 388, 2756, 20872, 180008, 1758672, 18937152, 221914944, 2832193008, 39039810912, 575502635808, 9100950684480, 152818028328960, 2717564023296000, 51129136369981440, 1012979833297735680, 21074454817487953920, 460035753479203184640
Offset: 0
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nmax = 21; CoefficientList[Series[Product[(1 + x^k/k)^2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A371493
Expansion of e.g.f. Product_{k>=2} (1 + x^k/k).
Original entry on oeis.org
1, 0, 1, 2, 6, 44, 210, 1644, 11088, 119664, 1034640, 12372480, 139629600, 1877722560, 25389131040, 395162832960, 6041860070400, 105872058754560, 1864694944465920, 35822359116149760, 705399920144640000, 15048474234019430400, 324762706938629836800, 7566557300438795366400
Offset: 0
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nmax = 23; CoefficientList[Series[Product[(1 + x^k/k), {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A371548
Expansion of e.g.f. Product_{k>=1} (1 + 2*x^k/k).
Original entry on oeis.org
1, 2, 2, 16, 44, 248, 2136, 13536, 116448, 1075392, 13066560, 136385280, 1811975040, 23777683200, 354509003520, 5632664970240, 93712140103680, 1631567291719680, 30968760551178240, 605247894280028160, 12515132360676556800, 274444506310599475200
Offset: 0
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nmax = 21; CoefficientList[Series[Product[(1 + 2 x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
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