A242446 -id:A242446 - 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

A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

Original entry on oeis.org

1, 1, 6, 57, 796, 15145, 374526, 11669665, 447595800, 20733553809, 1141067915290, 73552752257281, 5484203261135028, 467864288815609465, 45236104846954021014, 4915818294874879570305, 596044703812665607374256, 80118478395137652912476449, 11870487496575403846760198322

Offset: 0

Vaclav Kotesovec, Jun 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..283
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Cf. A000110, A031971, A072034, A242446, A337001, A337002, A354436.

Main diagonal of A337085.

Join[{1}, Table[n!*Sum[k^n/k!,{k,0,n}],{n,1,20}]]

a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-k*x))))) \\ Seiichi Manyama, Aug 23 2022

a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.
a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).
E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - Seiichi Manyama, Aug 23 2022
a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024
a(n) = Sum_{k=0..n} k^n*(n-k)!*binomial(n,k). - Ridouane Oudra, Jun 16 2025

a(0)=1 prepended by Seiichi Manyama, Aug 14 2020

A242449 a(n) = Sum_{k=0..n} C(n,k) * (2k+1)^(2n+1).

Original entry on oeis.org

1, 28, 3612, 1064480, 560632400, 462479403072, 550095467201728, 891290348282967040, 1887146395301619304704, 5058811707344107766328320, 16746136671945501439084657664, 67088193422344140016282100785152, 319900900946743851959321101768511488

Offset: 0

Vaclav Kotesovec, May 14 2014

Generally, for p>=1, a(n) = Sum_{k=0..n} C(n,k) * (p*k+1)^(p*n+1) is asymptotic to n^(p*n+1) * p^(p*n+1) * r^(p*n+3/2+1/p) / (sqrt(p+r-p*r) * exp(p*n) * (1-r)^(n+1/p)), where r = p/(p+LambertW(p*exp(-p))).

G. C. Greubel, Table of n, a(n) for n = 0..190

Cf. A242373, A220955, A242446, A072034, A007820, A221214, A213193, A195242.

Table[Sum[Binomial[n,k]*(2*k+1)^(2*n+1),{k,0,n}],{n,0,20}]

for(n=0,30, print1(sum(k=0,n, binomial(n,k)*(2*k+1)^(2*n+1)), ", ")) \\ G. C. Greubel, Nov 16 2017

a(n) ~ n^(2*n+1) * 2^(2*n+1) * r^(2*n+2) / (sqrt(2-r) * exp(2*n) * (1-r)^(n+1/2)), where r = 2/(2+LambertW(2*exp(-2))) = 0.901829091937052...

A323280 a(n) = Sum_{k=0..n} binomial(n,k) * k^(2*k).

Original entry on oeis.org

1, 2, 19, 781, 68553, 10100761, 2236373953, 693667946945, 286962262702657, 152652510206521921, 101513694573289791441, 82511051259976074269425, 80480313356721971865934369, 92773167329045961244649105633, 124768226258051318899374299271601

Offset: 0

Seiichi Manyama, Jan 12 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. A062206, A064570, A086331, A242446, A277454, A277456, A308490, A316747.

Table[1 + Sum[Binomial[n, k]*k^(2*k), {k, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, May 31 2019 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k^2*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022

a(n) ~ n^(2*n). - Vaclav Kotesovec, May 31 2019
From Seiichi Manyama, Jul 04 2022: (Start)
G.f.: Sum_{k>=0} (k^2 * x)^k/(1 - x)^(k+1).
E.g.f.: exp(x) * Sum_{k>=0} (k^2 * x)^k/k!. (End)

A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order.

Original entry on oeis.org

1, 2, 98, 11880, 2432430, 714249900, 275335499824, 131928199603200, 75727786603836510, 50713478000403718500, 38843740303576863755100, 33508462196084294380001040, 32157574295254903735909896240, 33990046387543889224733323929120

Offset: 0

Alois P. Heinz, May 28 2015

nonn

			a(0) = 1: the empty string.
a(1) = 2: ()(), (()).
a(2) = A000108(4) * (2^3-1) = 14*7 = 98.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A000108, A210029, A242446, A253180, A256254, A258426.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
a:= n-> add(A(2*n, n-i)*(-1)^i/((n-i)!*i!), i=0..n):
seq(a(n), n=0..15);

A[n_, k_] := A[n, k] = k^n CatalanNumber[n];
a[n_] := If[n==0, 1, Sum[A[2n, n-i] (-1)^i/((n-i)! i!), {i, 0, n}]];
a /@ Range[0, 15] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

a(n) = A253180(2n,n).
a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = -64/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 98.8248737517356857317..., c = 1/(2^(5/2) * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.0412044746356859529237459292541572856326... . - Vaclav Kotesovec, Jun 01 2015, updated Sep 27 2023
a(n) = A210029(n) * (4*n)! / (n! * (2*n)! * (2*n + 1)!), for n>0. - Vaclav Kotesovec, Sep 27 2023

A355470 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - k^3 * x)^(k+1).

Original entry on oeis.org

1, 1, 66, 21222, 18927560, 36030104000, 125486684755152, 722272396672485568, 6391048590559497227904, 82362961035803105954736768, 1482370265813455598541301007360, 36031982428595760278113744699088384, 1150873035676373345725887922070318410752

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A355469, A355473.

Cf. A072034, A242446, A355466.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^3*x)^k/(1-k^3*x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(1+sum(k=1, N, exp(k^3*x)*(k^3*x)^k/k!)))

a(n) = sum(k=0, n, k^(3*n)*binomial(n, k));

E.g.f.: Sum_{k>=0} exp(k^3 * x) * (k^3 * x)^k/k!.

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

A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

Original entry on oeis.org

1, 2, 19, 19879, 4297094601, 298028721578591321, 10314430386430205371442173873, 256923580889667562995278943476559835493321, 6277101737079381674883855772624745947410338680458857322625

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A072034, A242446, A355470.

Cf. A349886.

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, (k^k*x)^k/(1-k^k*x)^(k+1)))

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(k^k*x)*(k^k*x)^k/k!)))

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

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

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

A355468 Expansion of Sum_{k>=0} (k^2 * x/(1 - k^2 * x))^k.

Original entry on oeis.org

1, 1, 17, 858, 85988, 14318320, 3570592512, 1245401343760, 578840603221568, 345763649636940672, 258099498410703320960, 235426611021544158413824, 257654470061373320338925568, 333210260028337620911268462592

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..211

Cf. A195242, A242446, A249459.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x/(1-k^2*x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k^(2*n)*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k^(2*n) * binomial(n-1,k-1) for n > 0.

A381459 a(n) = (2n)! [x^(2*n)] cosh(x)^n.

Original entry on oeis.org

1, 1, 8, 183, 8320, 628805, 71172096, 11266376947, 2376282177536, 644092653605769, 218152097885716480, 90283850458537906511, 44828889635978905387008, 26302150870235970074916493, 18001952557737056033350615040, 14215240470695667525160827723915

Offset: 0

Seiichi Manyama, May 11 2025

nonn

Main diagonal of A326476.

Cf. A242446.

Table[(2*n)! * SeriesCoefficient[Cosh[x]^n, {x, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, May 11 2025 *)

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

a(n) = (1/2^n) * Sum_{k=0..n} (n-2*k)^(2*n) * binomial(n,k).

a(n) ~ c * ((1-2*r)^2 / (2 * r^r * (1-r)^(1-r)))^n * n^(2*n), where r = 0.015817782507793257357841601600685290637088885324182071456255... is the root of the equation (1-2*r)*(log(1-r) - log(r)) = 4 and c = 2*(1 - 2*r) / sqrt(1 + 4*r - 4*r^2) = 1.879106100687674868112932937483753439332007654254262530564... - Vaclav Kotesovec, May 11 2025, updated May 12 2025

cp's OEIS Frontend

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

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A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

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A242449 a(n) = Sum_{k=0..n} C(n,k) * (2*k+1)^(2*n+1).

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A323280 a(n) = Sum_{k=0..n} binomial(n,k) * k^(2*k).

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A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order.

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A355470 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - k^3 * x)^(k+1).

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A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

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A242449 a(n) = Sum_{k=0..n} C(n,k) * (2k+1)^(2n+1).

A381459 a(n) = (2n)! [x^(2*n)] cosh(x)^n.