A242817 -id:A242817 - cp's OEIS frontend

A337057 a(n) = exp(-n) * Sum_{k>=0} (k - n)^n * n^k / k!.

1, 0, 2, 3, 52, 255, 4146, 38766, 688584, 9685017, 195875110, 3655101703, 84872077500, 1955205893680, 51896551499898, 1412668946049315, 42475968202854160, 1328074354724554471, 44778480417250291566, 1577210136570598631318

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

nonn

Cf. A000296, A194689, A242817, A334242, A335867.

Table[n! SeriesCoefficient[Exp[n (Exp[x] - 1 - x)], {x, 0, n}], {n, 0, 19}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] (-n)^(n - k) BellB[k, n], {k, 0, n}], {n, 0, 19}]

a(n) = n! * [x^n] exp(n*(exp(x) - 1 - x)).

a(n) = Sum_{k=0..n} binomial(n,k) * (-n)^(n-k) * BellPolynomial_k(n).

A343263 a(0) = 1; a(n+1) = exp(-a(n)) * Sum_{k>=0} a(n)^k * k^n / k!.

Original entry on oeis.org

1, 1, 1, 2, 22, 301554, 2493675105669492542968967478

Offset: 0

Ilya Gutkovskiy, Apr 09 2021

nonn

The next term is too large to include.

Cf. A000110, A003659, A067691, A110386, A189233, A242817, A292860, A294373.

a[0] = 1; a[n_] := a[n] = Sum[StirlingS2[n - 1, k] a[n - 1]^k, {k, 0, n - 1}]; Table[a[n], {n, 0, 6}]
a[0] = 1; a[n_] := a[n] = BellB[n - 1, a[n - 1]]; Table[a[n], {n, 0, 6}]

a(0) = 1; a(n+1) = n! * [x^n] exp(a(n) * (exp(x) - 1)).

a(0) = 1; a(n+1) = Sum_{k=0..n} Stirling2(n,k) * a(n)^k.

cp's OEIS Frontend

A337057 a(n) = exp(-n) * Sum_{k>=0} (k - n)^n * n^k / k!.

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