A195242 -id:A195242 - cp's OEIS frontend

A053506 a(n) = (n-1)*n^(n-2).

0, 1, 6, 48, 500, 6480, 100842, 1835008, 38263752, 900000000, 23579476910, 681091006464, 21505924728444, 737020860878848, 27246730957031250, 1080863910568919040, 45798768824157052688, 2064472028642102280192, 98646963440126439346902, 4980736000000000000000000

Offset: 1

N. J. A. Sloane, Jan 15 2000

nonn

a(n) is the number of endofunctions f of [n] which interchange a pair a<->b and for all x in [n] some iterate f^k(x) = a. E.g., a(3) = 6: 1<->2<-3; 3->1<->2; 2<->3<-1; 1->2<->3; 1<->3<-2; 2->1<->3. - Len Smiley, Nov 27 2001
If offset is 0: right side of the binomial sum n-> sum( i^(i-1) * (n-i+1)^(n-i)*binomial(n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
a(n) is the number of birooted labeled trees on n nodes in which the two root nodes are adjacent. - N. J. A. Sloane, May 01 2018
a(n) is the number of ways to partition the complete graph K_n into two components and choose an arborescence on each component. - Harry Richman, May 11 2022

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.36)
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.

G. C. Greubel, Table of n, a(n) for n = 1..250
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Eric Weisstein's World of Mathematics, Graph Edge

Cf. A000169, A000312, A053507, A053508, A053509, A083483, A195242.

Cf. A001865 which is the sum of A000169 + A053506 + A065513 + A065888 + ...

List([1..20], n-> (n-1)*n^(n-2)) # G. C. Greubel, May 15 2019

[(n-1)*n^(n-2): n in [1..20]]; // G. C. Greubel, May 15 2019

Mathematica
```
Table[(n-1)*n^(n-2), {n,20}]
```

vector(20, n, (n-1)*n^(n-2)) \\ G. C. Greubel, Jan 18 2017

[(n-1)*n^(n-2) for n in (1..20)] # G. C. Greubel, May 15 2019

E.g.f.: LambertW(-x)^2/2. - Vladeta Jovovic, Apr 07 2001
E.g.f. if offset 0: W(-x)^2/((1+W(-x))*x), W(x) Lambert's function (principal branch).
The sequence 1, 1, 6, 48, ... satisfies a(n) = (n*(n+1)^n + 0^n)/(n+1); it is the main diagonal of A085388. - Paul Barry, Jun 30 2003
a(n) = Sum_{i=1..n-1} binomial(n-1,i-1)*i^(i-2)*(n-i)^(n-i). - Dmitry Kruchinin, Oct 28 2013
If offset = 0 and a(0) = 1 then a(n) = Sum_{k=0..n} (-1)^(n-k)* binomial(-k,-n)*n^k (cf. A195242). - Peter Luschny, Apr 11 2016

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...

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.

A358740 Expansion of Sum_{k>=0} k! * ( k * x/(1 - k*x) )^k.

Original entry on oeis.org

1, 1, 9, 195, 7699, 482309, 43994741, 5508667927, 906931827831, 189998213001033, 49359340639141993, 15573690455085072011, 5866304418414451865723, 2600416934781350100016717, 1340037064604153376788884701, 794358527033920600533985973631

Offset: 0

Seiichi Manyama, Nov 29 2022

nonn

Cf. A358738, A358741.

Cf. A195242.

nmax = 20; CoefficientList[1 + Series[Sum[k! * (k * x/(1 - k*x))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)

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

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

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

a(n) ~ exp(exp(-1)) * n! * n^n. - Vaclav Kotesovec, Feb 18 2023

A360818 Expansion of Sum_{k>=0} ( (kx)^2 / (1 - kx) )^k.

Original entry on oeis.org

1, 0, 1, 1, 17, 65, 922, 7074, 106183, 1248479, 21144289, 331763177, 6441011484, 124904347404, 2773880604749, 63538143151589, 1600211849569585, 42076439530000297, 1189408501356380558, 35214128238218917974, 1106088535644470694779

Offset: 0

Seiichi Manyama, Feb 21 2023

nonn

Cf. A195242, A360819.

Cf. A360708.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, ((k*x)^2/(1-k*x))^k))

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

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

A360819 Expansion of Sum_{k>=0} ( (kx)^3 / (1 - kx) )^k.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 65, 257, 769, 21732, 182268, 1075171, 22120299, 292415838, 2784944366, 52394511682, 914813711338, 12411977351379, 240868108545883, 5024364548461861, 88977315031536205, 1888119425325238979, 44744897995532996819, 971263427084750362992

Offset: 0

Seiichi Manyama, Feb 21 2023

nonn

Cf. A195242, A360818.

Cf. A360709.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, ((k*x)^3/(1-k*x))^k))

a(n) = sum(k=0, n\3, k^n*binomial(n-2*k-1, n-3*k));

a(n) = Sum_{k=0..floor(n/3)} k^n * binomial(n-2*k-1,n-3*k).

A360832 Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^2) )^k.

Original entry on oeis.org

1, 1, 4, 28, 288, 3855, 63232, 1227291, 27511296, 699389444, 19880700928, 624817997477, 21512488648704, 805233062024021, 32556682898653184, 1413981749074790444, 65652661019642560512, 3245240681196968168619, 170146759140135777861632

Offset: 0

Seiichi Manyama, Feb 22 2023

nonn

Cf. A195242, A360833.

Cf. A338661, A360810, A360834.

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

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

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

A360833 Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^3) )^k.

Original entry on oeis.org

1, 1, 4, 27, 257, 3189, 48843, 889080, 18731109, 448004763, 11987812504, 354763577414, 11503684020051, 405589341060930, 15447798292502206, 632069580794524857, 27649951709582591394, 1287748889361331630661, 63616184683123273364961

Offset: 0

Seiichi Manyama, Feb 22 2023

nonn

Cf. A195242, A360832.

Cf. A338661, A360811, A360835.

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

a(n) = sum(k=0, n\3, (n-3*k)^n*binomial(n-2*k-1, k));

a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^n * binomial(n-2*k-1,k).

cp's OEIS Frontend

A053506 a(n) = (n-1)*n^(n-2).

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

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

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PARI

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A358740 Expansion of Sum_{k>=0} k! * ( k * x/(1 - k*x) )^k.

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Mathematica

PARI

PARI

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A360818 Expansion of Sum_{k>=0} ( (k*x)^2 / (1 - k*x) )^k.

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PARI

PARI

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A360819 Expansion of Sum_{k>=0} ( (k*x)^3 / (1 - k*x) )^k.

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PARI

PARI

Formula

A360832 Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^2) )^k.

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PARI

PARI

Formula

A360833 Expansion of Sum_{k>=0} ( k * x / (1 - (k * x)^3) )^k.

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Author

Keywords

Crossrefs

Programs

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

A360818 Expansion of Sum_{k>=0} ( (kx)^2 / (1 - kx) )^k.

A360819 Expansion of Sum_{k>=0} ( (kx)^3 / (1 - kx) )^k.