A347434 -id:A347434 - cp's OEIS frontend

A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

1, 0, 1, 4, 14, 66, 397, 2626, 18797, 148238, 1281134, 11943790, 118998365, 1262189748, 14203022537, 168835162632, 2111832477426, 27708387132906, 380355066174121, 5449577398256414, 81316095965242989, 1261149374033472626, 20293627142875917978, 338263983223664609198

Offset: 0

Ilya Gutkovskiy, Sep 02 2021

nonn

Exponential transform of A000295.

Cf. A000295, A000296, A003725, A055882, A143405, A347434, A347435.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021

nmax = 23; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000295(k) * a(n-k).
a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A003725(k) * A143405(n-k).
a(n) ~ n^(n + 1/2) * (exp(exp(r)*(exp(r) - r - 1) - r/2 - n) / (r^(n + 1/2) * sqrt(2*exp(r)*(1 + 2*r) - (2 + r*(4 + r))))), where r = LambertW(n)/2 + (4 + LambertW(n)) * LambertW(n)^(3/2) / (8 * sqrt(n) * (1 + LambertW(n))). - Vaclav Kotesovec, Jul 07 2022

A347435 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6) ).

Original entry on oeis.org

1, 0, 0, 0, 1, 6, 22, 64, 198, 1138, 10004, 83920, 617993, 4226028, 30103686, 251883012, 2490287821, 26456763078, 281404300348, 2966101610920, 31877462564554, 362624252399566, 4437794875670072, 57612897938229380, 773900876490016325, 10599854900351622752

Offset: 0

Ilya Gutkovskiy, Sep 02 2021

nonn

Exponential transform of A002663.

Cf. A002663, A055882, A057837, A143405, A347432, A347434.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j^3/6-5*j/6-1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x - x^2/2 - x^3/6)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - 1 - k (k^2 + 5)/6) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002663(k) * a(n-k).

cp's OEIS Frontend

A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

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