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]!
A294470
E.g.f.: Product_{k>0} (1-x^k/k)^k.
Original entry on oeis.org
1, -1, -2, 0, 6, 90, 240, 2100, 1680, -28560, -682080, -7835520, -134497440, -1059458400, -13047914880, -138646347840, -1130671301760, 3558932657280, 93341485036800, 7903311136680960, 149238605943947520, 3997338754242113280, 86934078099901992960
Offset: 0
A345752
E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k / k).
Original entry on oeis.org
1, -1, -2, -3, 0, 55, 572, 4865, 40912, 351675, 2978196, 23418373, 148849544, 84185855, -27459134420, -881482705719, -21652972750464, -487503384038525, -10785437160748156, -242968902040697011, -5627949704687484872, -133358411031825299385
Offset: 0
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max = 21; Range[0, max]! * CoefficientList[Series[Product[1 - (Exp[x] - 1)^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1-(exp(x)-1)^k/k)))
A346312
Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} (1 - x^n / n^2).
Original entry on oeis.org
1, -1, -1, 5, 28, 724, 36, 220716, -1255680, 110979072, 2530310400, 1193835283200, -24457819622400, 21656019855744000, 899271273253248000, 474367063601421849600, 45822442913828595302400, 28365278076547150440038400, 2614371018285307258994688000
Offset: 0
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nmax = 18; CoefficientList[Series[Product[(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[(Binomial[n, k] k!)^2 Sum[1/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
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