A291979
a(n) = (-1)^n*n!*[x^n] exp(-x)/(1 + log(1+x)).
Original entry on oeis.org
1, 2, 6, 27, 167, 1310, 12394, 137053, 1733325, 24670114, 390204086, 6789564639, 128884276179, 2650516064222, 58701784670138, 1392959655437473, 35257885037803417, 948208649740610466, 27000743345935785670, 811575543670852269347, 25677856392014665436799
Offset: 0
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a_list := proc(n) exp(-x)/(1 + log(1+x)): series(%, x, n+1):
seq((-1)^k*k!*coeff(%, x, k), k=0..n) end: a_list(20);
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nmax = 20; CoefficientList[Series[E^(-x)/(1 + Log[1+x]), {x, 0, nmax}], x] * Range[0, nmax]! * (-1)^Range[0, nmax] (* Vaclav Kotesovec, Sep 18 2017 *)
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N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+log(1-x)))) \\ Seiichi Manyama, Oct 20 2021
A343707
a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).
Original entry on oeis.org
1, 3, 15, 113, 1145, 14539, 221663, 3943281, 80173345, 1833831619, 46606646175, 1302954958689, 39737420405753, 1312901360002283, 46714233470065999, 1780859204826798401, 72416689888874547969, 3128792006916853876291, 143132514626658326870767, 6911638338982428907738641
Offset: 0
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a[n_] := a[n] = 1 + 2 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x]/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
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N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+2*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021
A343710
a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).
Original entry on oeis.org
1, 5, 45, 609, 11009, 248837, 6749629, 213596401, 7725031521, 314310704101, 14209394894765, 706617979262049, 38333841625642785, 2252901018519028901, 142589176837851349757, 9669282207517755852721, 699408060608904410296897, 53752166013267632536864581, 4374061543586452325644329133
Offset: 0
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a[n_] := a[n] = 1 + 4 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[x]/(1 + 4 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
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N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+4*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021
A368445
Expansion of e.g.f. exp(x) / (1 + log(1 - 3*x)).
Original entry on oeis.org
1, 4, 34, 469, 8815, 208348, 5922118, 196568419, 7459854973, 318560689324, 15116763184978, 789119869380577, 44939583072146251, 2772582488089509028, 184216538154508055062, 13114092114632287359919, 995813104288130697683065, 80342826520464644566291828
Offset: 0
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With[{nn=20},CoefficientList[Series[Exp[x]/(1+Log[1-3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 07 2025 *)
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a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=1, i, 3^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;
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