A343928 -id:A343928 - cp's OEIS frontend

A343899 a(n) = Sum_{k=0..n} (k!)^k * binomial(n,k).

1, 2, 7, 232, 332669, 24884861086, 139314218808181027, 82606412229102532926819812, 6984964247802365417561163907914436537, 109110688415634181158572146813823590758078301022074, 395940866122426284350759726810156652343313286283891529199276099071

Offset: 0

Seiichi Manyama, May 03 2021

nonn

Binomial transform of (n!)^n.

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000522, A036740, A046662, A086331, A343898, A343900, A343928.

a[n_] := Sum[(k!)^k * Binomial[n, k], {k, 0, n}]; Array[a, 11, 0] (* Amiram Eldar, May 05 2021 *)

a(n) = sum(k=0, n, k!^k*binomial(n, k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k!*x)^k/(1-x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, k!^(k-1)*x^k)))

G.f.: Sum_{k>=0} (k! * x)^k/(1 - x)^(k+1).

E.g.f.: exp(x) * Sum_{k>=0} (k!)^(k-1) * x^k.

A343929 a(n) = Sum_{k=0..n} (k!)^(n+1) * binomial(n,k).

Original entry on oeis.org

1, 2, 11, 1348, 7993925, 2986939982086, 100308280020162672007, 416336818263472141683094788104, 281633775231427434285800695714399092181001, 39594086714441777969538839399390619086007952991080833034

Offset: 0

Seiichi Manyama, May 04 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..29

Cf. A046662, A343898, A343900, A343928.

a[n_] := Sum[(k!)^(n+1) * Binomial[n, k], {k, 0, n} ]; Array[a, 10, 0] (* Amiram Eldar, May 04 2021 *)

a(n) = sum(k=0, n, k!^(n+1)*binomial(n, k));

a(n) = [x^n] Sum_{k>=0} (k!)^(n+1) * x^k/(1 - x)^(k+1).

a(n) = n! * [x^n] exp(x) * Sum_{k>=0} (k!)^n * x^k.

cp's OEIS Frontend

A343899 a(n) = Sum_{k=0..n} (k!)^k * binomial(n,k).

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A343929 a(n) = Sum_{k=0..n} (k!)^(n+1) * binomial(n,k).

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Mathematica

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