A331797 -id:A331797 - cp's OEIS frontend

A331796 E.g.f.: (exp(x) - 1) * exp(1 - exp(x)) / (2 - exp(x)).

0, 1, 1, 4, 27, 201, 1730, 17403, 200753, 2607034, 37614509, 596935373, 10334325760, 193820393781, 3914731176005, 84716449797164, 1955520065429447, 47960724916860501, 1245468600257306394, 34139796085144434199, 985066290121984334613

Offset: 0

Ilya Gutkovskiy, Jan 26 2020

nonn

Stirling transform of A000240.

Alois P. Heinz, Table of n, a(n) for n = 0..424

Cf. A000240, A000587, A000670, A064898, A331797.

g:= proc(n) option remember;
     `if`(n=0, 0, n*(g(n-1)-(-1)^n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 23 2023

nmax = 20; CoefficientList[Series[(Exp[x] - 1) Exp[1 - Exp[x]]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
A000240[n_] := n! Sum[(-1)^k/k!, {k, 0, n - 1}]; a[n_] := Sum[StirlingS2[n, k] A000240[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, -k, 0] BellB[n - k, -1], {k, 1, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A000240(k).
a(n) = Sum_{k=1..n} binomial(n,k) * A000670(k) * A000587(n-k).
a(n) ~ n! * exp(-1) / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020

A331798 E.g.f.: -log(1 - x) / ((1 - x) * (1 + log(1 - x))).

Original entry on oeis.org

0, 1, 5, 29, 204, 1714, 16862, 190826, 2447512, 35136696, 558727872, 9754239648, 185546362416, 3820734689472, 84687887312688, 2010622152615504, 50908186083448320, 1369376758488222336, 38998680958184088960, 1172297572938013827456, 37092793335394301708544

Offset: 0

Ilya Gutkovskiy, Jan 26 2020

nonn

Cf. A000254, A001008, A002805, A003713, A007526, A007840, A215916, A331797.

nmax = 20; CoefficientList[Series[-Log[1 - x]/((1 - x) (1 + Log[1 - x])), {x, 0, nmax}], x] Range[0, nmax]!
A007526[n_] := n! Sum[1/k!, {k, 0, n - 1}]; a[n_] := Sum[Abs[StirlingS1[n, k]] A007526[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
A007840[n_] := Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}]; a[n_] := Sum[Binomial[n, k] k! HarmonicNumber[k] A007840[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

a(n) = Sum_{k=0..n} = |Stirling1(n,k)| * A007526(k).
a(n) = Sum_{k=1..n} binomial(n,k) * k! * H(k) * A007840(n-k), where H(k) is the k-th harmonic number.
a(n) ~ n! / (1 - exp(-1))^(n+1). - Vaclav Kotesovec, Jan 26 2020

cp's OEIS Frontend

A331796 E.g.f.: (exp(x) - 1) * exp(1 - exp(x)) / (2 - exp(x)).

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