A294494
E.g.f.: 1/Product_{k>0} (1+x^k/k!)^k.
Original entry on oeis.org
1, -1, 0, -3, 26, -75, 324, -3535, 30988, -242025, 2245820, -26847381, 339741984, -4205748547, 57094691822, -883946426805, 14358210544304, -239959114870689, 4286519116236900, -82194727064059645, 1650577120959962440, -34495065863164195611
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, May 12 2018 *)
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N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1+x^k/k!)^k)))
A319219
Expansion of e.g.f. Product_{k>=1} 1/(1 + x^k/(k - 1)!).
Original entry on oeis.org
1, -1, 0, -3, 32, -105, 204, -3325, 52408, -376425, 1304180, -25766301, 659066484, -6675505837, 30765540974, -893416597515, 29169795361424, -380344619169729, 2379504317523300, -84225906785770525, 3388223174832010540, -55107296201168047221, 422923168260105913070
Offset: 0
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seq(n!*coeff(series(mul(1/(1 + x^k/(k - 1)!),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
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nmax = 22; CoefficientList[Series[Product[1/(1 + x^k/(k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(k (-(j - 1)!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[d (-(d - 1)!)^(-k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]
A346315
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, 28, 483, 11976, 423660, 20801775, 1337182819, 108259612048, 10814058518328, 1308659192928495, 188498906179378476, 31855351764833425895, 6243218508505581436249, 1404734813476218805338303, 359618310105650201828166499, 103929494668760259335327432160
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 k Sum[(-1)^d/(d ((k/d)!)^(2 d)), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
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