A343685 -id:A343685 - cp's OEIS frontend

A343686 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

1, 4, 33, 410, 6796, 140824, 3501782, 101589732, 3368237928, 125634319104, 5206805098752, 237370661584704, 11805144854303760, 636030155604374400, 36903603627294958416, 2294156656214759133024, 152126925169297299197184, 10718105879980375520103936, 799564645068022035991527552

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Cf. A007840, A052820, A053486, A343685, A343687.

a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(1 - 3 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 - 3*x + log(1 - x)).

a(n) ~ n! / ((3/c + 2 - c) * (1 - c/3)^n), where c = LambertW(3*exp(2)) = 2.2761339297716461777892556270138... - Vaclav Kotesovec, Apr 26 2021

A343687 a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 5, 51, 782, 15992, 408814, 12541010, 448834728, 18358297416, 844755218400, 43190363326992, 2429044756967520, 149029669269441456, 9905401062535389072, 709016063545908259248, 54375505616232613595904, 4448148376192382963462400, 386619861956492109750650496, 35580548688887294090357622912

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Cf. A007840, A052820, A053487, A343685, A343686.

a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(1 - 4 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 - 4*x + log(1 - x)).

a(n) ~ n! / ((4/c + 3 - c) * (1 - c/4)^n), where c = LambertW(4*exp(3)) = 3.2176447220005493578369738... - Vaclav Kotesovec, Apr 26 2021

cp's OEIS Frontend

A343686 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

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