A343929 -id:A343929 - cp's OEIS frontend

A343900 a(n) = Sum_{k=0..n} (k!)^(k+1) * binomial(n,k).

1, 2, 11, 1324, 7967861, 2986023826166, 100306147958903465407, 416336313421816733159702737376, 281633758448076539969292901914477101456489, 39594086612245054028213574779019294652734771094507399786

Offset: 0

Seiichi Manyama, May 03 2021

nonn

Binomial transform of (n!)^(n+1).

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

Cf. A000522, A036740, A046662, A091868, A343898, A343899, A343929.

a[n_] := Sum[(k!)^(k+1) * Binomial[n, k], {k, 0, n}]; Array[a, 10, 0] (* Amiram Eldar, May 05 2021 *)

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

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

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

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

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

A343928 a(n) = Sum_{k=0..n} (k!)^n * binomial(n,k).

Original entry on oeis.org

1, 2, 7, 244, 337061, 24923091206, 139331988275478727, 82607113404338664216300296, 6984967577834038055008791270166057993, 109110690950275218023122492287310115968068596613130, 395940866518366059877297056617763923418318903997411043997258716171

Offset: 0

Seiichi Manyama, May 04 2021

nonn

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

Cf. A000522, A046662, A072034, A336247, A339311, A343898, A343899, A343929.

a[n_] := Sum[(k!)^n * Binomial[n, k], {k, 0, n} ]; Array[a, 11, 0] (* Amiram Eldar, May 04 2021 *)

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

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

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

cp's OEIS Frontend

A343900 a(n) = Sum_{k=0..n} (k!)^(k+1) * binomial(n,k).

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A343928 a(n) = Sum_{k=0..n} (k!)^n * binomial(n,k).

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