A305199
Expansion of e.g.f. Product_{k>=1} (1 + x^k/k)/(1 - x^k/k).
Original entry on oeis.org
1, 2, 6, 28, 152, 1008, 7756, 67688, 659424, 7123776, 84154224, 1079913888, 14962632384, 222447507072, 3531920599008, 59664827178048, 1067975819206656, 20192760528611328, 402169396496004864, 8414121277765679616, 184498963978904644608, 4231186653661629843456
Offset: 0
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a:=series(mul((1+x^k/k)/(1-x^k/k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
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nmax = 21; CoefficientList[Series[Product[(1 + x^k/k)/(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(1 + (-1)^(k + 1)) x^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A294471
E.g.f.: 1/Product_{k>0} (1+x^k/k)^k.
Original entry on oeis.org
1, -1, 0, -6, 18, -90, 660, -3360, 47880, -293160, 4277280, -36424080, 575190000, -6745218480, 101911249440, -1628086299840, 24861230634240, -484979925830400, 7629427896330240, -176975913961566720, 3036472694482106880, -77953392499390087680
Offset: 0
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With[{nn=30},CoefficientList[Series[1/Product[(1+x^k/k)^k,{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 30 2021 *)
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N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1+x^k/k)^k)))
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}]
A346313
Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} 1 / (1 + (-x)^n / n^2).
Original entry on oeis.org
1, 1, 3, 31, 496, 12576, 444736, 22056448, 1406058816, 114618828096, 11405077216704, 1385889578069184, 198961869847145472, 33725910553646229504, 6594186368339077238784, 1487133154121568112705536, 379990326228614750079369216, 110013397755650063836228435968
Offset: 0
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nmax = 17; CoefficientList[Series[Product[1/(1 + (-x)^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[(-1)^k (Binomial[n, k] k!)^2 Sum[(-1)^d/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
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