A336828 -id:A336828 - cp's OEIS frontend

A072034 a(n) = Sum_{k=0..n} binomial(n,k)*k^n.

1, 1, 6, 54, 680, 11000, 217392, 5076400, 136761984, 4175432064, 142469423360, 5372711277824, 221903307604992, 9961821300640768, 482982946946734080, 25150966159083264000, 1400031335107317628928, 82960293298087664648192

Offset: 0

Karol A. Penson, Jun 07 2002

nonn

The number of functions from {1,2,...,n} into a subset of {1,2,...,n} summed over all subsets. - Geoffrey Critzer, Sep 16 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

Cf. A088789, A242446, A256016, A336828, A341815.

seq(add(binomial(n,k)*k^n,k=0..n),n=0..17); # Peter Luschny, Jun 09 2015

Table[Sum[Binomial[n,k]k^n,{k,0,n}],{n,1,20}] (* Geoffrey Critzer, Sep 16 2012 *)

x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 10 2017

a(n) = sum(k=0, n, binomial(n,k)*k^n); \\ Michel Marcus, Nov 10 2017

E.g.f.: 1/(1+LambertW(-x*exp(x))). - Vladeta Jovovic, Mar 29 2008
a(n) ~ (n/(e*LambertW(1/e)))^n/sqrt(1+LambertW(1/e)). - Vaclav Kotesovec, Nov 26 2012
O.g.f.: Sum_{n>=0} n^n * x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018

Offset set to 0 and a(0) = 1 prepended by Peter Luschny, Jun 09 2015

E.g.f. edited to include a(0)=1 by Robert Israel, Jun 09 2015

A341815 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k^n.

Original entry on oeis.org

1, 1, 12, 270, 8960, 406250, 23293872, 1617774592, 132075970560, 12397121784954, 1315528361642000, 155743010418063860, 20351866171905066240, 2909818652684404979440, 451849287590990124662400, 75730203998219999637000000, 13625593688459657260608782336, 2619521322904712777031960349850

Offset: 0

Vaclav Kotesovec, Feb 20 2021

nonn

For m>0, Sum_{k=0..n} binomial(n,k)^m * k^n ~ c(m) * d^n * n! / n^(m/2), where d = (1 + m*LambertW(exp(-1/m)/m))^(m-1) / (m^m * LambertW(exp(-1/m)/m)^m) and c(m) is a constant independent of n.

Cf. A072034, A336828.

Join[{1}, Table[Sum[k^n * Binomial[n, k]^3, {k, 0, n}], {n, 1, 20}]]

a(n) ~ c * d^n * n! / n^(3/2), where d = (1 + 3*LambertW(exp(-1/3)/3))^2 / (27 * LambertW(exp(-1/3)/3)^3) = 12.3645613141726293982008517178673172577947617775... and c = 0.143687082995832067469009730530027989920523409582173778129054767279...

A341185 a(n) = Sum_{k=0..n} k^n * k! * binomial(n,k)^2.

Original entry on oeis.org

1, 1, 12, 315, 15088, 1141625, 124989156, 18659050795, 3638892086208, 897534389449809, 272981684150035300, 100316132701760094251, 43802068733570039425776, 22409162143775383385763913, 13274030650412266312507931652

Offset: 0

Seiichi Manyama, Feb 06 2021

nonn

Cf. A002720, A332408, A336828, A341197.

a[0] = 1; a[n_] := Sum[k^n * k! * Binomial[n, k]^2, {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 06 2021 *)

a(n) = sum(k=0, n, k^n*k!*binomial(n, k)^2);

A336955 a(n) = Sum_{k=0..n} k^k * binomial(n, k)^2.

Original entry on oeis.org

1, 2, 9, 73, 849, 12651, 228493, 4836301, 117204545, 3196763983, 96842596701, 3224356269597, 116981406934417, 4591908332288837, 193846634326107701, 8755364023207809301, 421214258699748184321, 21500563181275847468503, 1160430732790051008442141, 66020998289431649938896445

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

Cf. A037966, A037972, A086331, A329913, A329444, A336828.

a[n_] := 1 + Sum[k^k * Binomial[n, k]^2, {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Aug 09 2020 *)

{a(n) = sum(k=0, n, k^k*binomial(n, k)^2)}

cp's OEIS Frontend

A072034 a(n) = Sum_{k=0..n} binomial(n,k)*k^n.

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A341815 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k^n.

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A341185 a(n) = Sum_{k=0..n} k^n * k! * binomial(n,k)^2.

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A336955 a(n) = Sum_{k=0..n} k^k * binomial(n, k)^2.

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