A007778 -id:A007778 - cp's OEIS frontend

A008787 a(n) = (n + 6)^n.

1, 7, 64, 729, 10000, 161051, 2985984, 62748517, 1475789056, 38443359375, 1099511627776, 34271896307633, 1156831381426176, 42052983462257059, 1638400000000000000, 68122318582951682301, 3011361496339065143296

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, this sequence, A008788, A008789, A008790, A008791.

List([0..20], n-> (n+6)^n); # G. C. Greubel, Sep 11 2019

[(n+6)^n: n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

Maple
```
a:= n-> (n+6)^n: seq(a(n), n=0..20);
```

Table[(n+6)^n,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n+5)^(n-1)) \\ G. C. Greubel, Sep 11 2019

[(n+6)^n for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.(x) for b(n) = n^(n-6) = a(n-6): T - (31/32)*T^2 + (575/1296)*T^3 - (415/3456)*T^4 + (137/7200)*T^5 - (1/720)*T^6; where T=T(x) is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
E.g.f.: LambertW(-x)^6/(x^6*(1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003
E.g.f.: (1/5)*d/dx(LambertW(-x)/(-x))^5. - Wolfdieter Lang, Oct 25 2022

A008790 a(n) = n^(n+4).

Original entry on oeis.org

0, 1, 64, 2187, 65536, 1953125, 60466176, 1977326743, 68719476736, 2541865828329, 100000000000000, 4177248169415651, 184884258895036416, 8650415919381337933, 426878854210636742656, 22168378200531005859375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A000169, A000272, A000312, A007778, A007830, A008785, A008786, A008787, A008788, A008789, A008791.

List([0..20], n-> n^(n+4)); # G. C. Greubel, Sep 11 2019

[n^(n+4): n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

a:=n->mul(n,k=-3..n):seq(a(n),n=0..20); # Zerinvary Lajos, Jan 26 2008

Table[n^(n+4),{n,0,20}](* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

vector(20, n, (n-1)^(n+3)) \\ G. C. Greubel, Sep 11 2019

[n^(n+4) for n in (0..20)] # G. C. Greubel, Sep 11 2019

E.g.f.: T*(1 +22*T +58*T^2 +24*T^3)*(1-T)^(-9); where T is Euler's tree function (see A000169). - Len Smiley, Nov 17 2001
See A008517 and A134991 for similar e.g.f.s and diagonals of A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d^4/dx^4 {x^4/(T(x)^4*(1-T(x)))}, where T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012

A283498 a(n) = Sum_{d|n} d^(d+1).

Original entry on oeis.org

1, 9, 82, 1033, 15626, 280026, 5764802, 134218761, 3486784483, 100000015634, 3138428376722, 106993205660122, 3937376385699290, 155568095563577034, 6568408355712906332, 295147905179487044617, 14063084452067724991010, 708235345355341163422059, 37589973457545958193355602

Offset: 1

Seiichi Manyama, Mar 09 2017

nonn

			a(6) = 1^2 + 2^3 + 3^4 + 6^7 = 280026.

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A007778, A062796 (Sum_{d|n} d^d).

f[n_] := Block[{d = Divisors[n]}, Total[d^(d + 1)]]; Array[f, 19] (* Robert G. Wilson v, Mar 10 2017 *)

a(n) = sumdiv(n, d, d^(d+1)); \\ Michel Marcus, Mar 09 2017

from sympy import divisors
def A283498(n): return sum(d**(d+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 19 2022

From Ilya Gutkovskiy, May 06 2017: (Start)
G.f.: Sum_{k>=1} k^(k+1)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^k)) = Sum_{n>=1} a(n)*x^n/n. (End)

More terms from Michel Marcus, Mar 09 2017

A085606 a(n) = (n-1)^n - 1.

Original entry on oeis.org

0, -1, 0, 7, 80, 1023, 15624, 279935, 5764800, 134217727, 3486784400, 99999999999, 3138428376720, 106993205379071, 3937376385699288, 155568095557812223, 6568408355712890624, 295147905179352825855, 14063084452067724991008, 708235345355337676357631

Offset: 0

Lekraj Beedassy, Jul 07 2003

sign

Sequence relates to the "monkey and coconut problem"(A014293) giving the number of coconuts received by each of the n sailors from the ultimate equitable distribution the next day.
From Alexander Adamchuk, Nov 13 2006: (Start)
4n^2 divides a(2n).
Odd prime p divides a(p-1).
8p^2 divides a(2p) for an odd prime p.
32p^4 divides a(2p^2) for an odd prime p.
64p^8 divides a(2p^4) for an odd prime p.
p^3 divides a(p^3+2) for prime p.
p divides a((p-1)/2) for prime p in A157437.
p^2 divides a((p-1)/2) for prime p = {5,127,607}. (End)

Cf. A014293, A007778, A065440, A037205, A157437.

Table[(n-1)^n-1,{n,0,30}] (* Alexander Adamchuk, Nov 13 2006 *)

a(n) = A065440(n) - 1.

More terms from Ray Chandler, Nov 10 2003

A110567 a(n) = n^(n+1) + 1.

Original entry on oeis.org

1, 2, 9, 82, 1025, 15626, 279937, 5764802, 134217729, 3486784402, 100000000001, 3138428376722, 106993205379073, 3937376385699290, 155568095557812225, 6568408355712890626, 295147905179352825857, 14063084452067724991010

Offset: 0

Jonathan Vos Post, Sep 12 2005

For n >= 2, a(n) = the n-th positive integer such that a(n) (base n) has a block of exactly n consecutive zeros.
Comments from Alexander Adamchuk, Nov 12 2006 (Start)
(2n+1)^2 divides a(2n). a(2n)/(2n+1)^2 = {1,1,41,5713,1657009,826446281,633095889817,691413758034721,...} = A081215(2n).
p divides a(p-1) for prime p. a(p-1)/p = {1,3,205,39991,9090909091,8230246567621,...} = A081209(p-1) = A076951(p-1).
p^2 divides a(p-1) for an odd prime p. a(p-1)/p^2 = {1,41,5713,826446281,633095889817,1021273028302258913,1961870762757168078553, 14199269001914612973017444081,...} = A081215(p-1).
Prime p divides a((p-3)/2) for p = {13,17,19,23,37,41,43,47,61,67,71,89, 109,113,137,139,157,163,167,181,191,...}.
Prime p divides a((p-5)/4) for p = {29,41,61,89,229,241,281,349,421,509,601,641,661,701,709,769,809,821,881,...} = A107218(n) Primes of the form 4x^2+25y^2.
Prime p divides a((p-7)/6) for p = {79,109,127,151,313,421,541,601,613,751,757,787,...}.
Prime p divides a((p-9)/8) for p = {41,337,401,521,569,577,601,857,929,937,953,977,...} A subset of A007519(n) Primes of form 8n+1.
Prime p divides a((p-11)/10) for p = {41,181,331,601,761,1021,1151,1231,1801,...}.
Prime p divides a((p-13)/12) for p = {313,337,433,1621,1873,1993,2161,2677,2833,...}. (End)

			Examples illustrating the Comment:
a(2) = 9 because the first positive integer (base 2) with a block of 2 consecutive zeros is 100 (base 2) = 4, and the 2nd is 1001 (base 2) = 9 = 1 + 2^3.
a(3) = 82 because the first positive integer (base 3) with a block of 3 consecutive zeros is 1000 (base 3) = 81, the 2nd is 2000 (base 3) = 54 and the 3rd is 10001 (base 3) = 82 = 1 + 3^4.
a(4) = 1025 because the first positive integer (base 4) with a block of 4 consecutive zeros is 10000 (base 4) = 256, the 2nd is 20000 (base 4) = 512, the 3rd is 30000 (base 4) = 768 and the 4th 100001 (base 4) = 1025 = 1 + 4^5. and the 2nd is 1001 (base 2) = 9 = 1 + 2^3.

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A007778: n^(n+1); A000312: n^n; A014566: Sierpinski numbers of the first kind: n^n + 1.

Cf also A081209, A076951, A081215. A110567.

[n^(n+1) + 1: n in [0..25]]; // G. C. Greubel, Oct 16 2017

Table[n^(n+1)+1,{n,0,30}] (* Harvey P. Dale, Oct 30 2015 *)

for(n=0,25, print1(1 + n^(n+1), ", ")) \\ G. C. Greubel, Aug 31 2017

a(n) = A007778(n) + 1.

a(n) = A110567(n) for n > 1. - Georg Fischer, Oct 20 2018

Entry revised by N. J. A. Sloane, Oct 20 2018 at the suggestion of Georg Fischer.

A084363 a(n) = n^(n+1) - (n-1)^n.

Original entry on oeis.org

1, 7, 73, 943, 14601, 264311, 5484865, 128452927, 3352566673, 96513215599, 3038428376721, 103854777002351, 3830383180320217, 151630719172112935, 6412840260155078401, 288579496823639935231, 13767936546888372165153

Offset: 1

Amarnath Murthy, May 27 2003

n-th partial arithmetic mean is n^n.

Cf. A000312, A007778.

Table[n^(n+1)-(n-1)^n, {n,25}] (* Vladimir Joseph Stephan Orlovsky, Dec 29 2010 *)

A084363[n]:=n^(n+1)-(n-1)^n$
makelist(A084363[n],n,1,30); /* Martin Ettl, Oct 29 2012 */

for(n=1,17,print1(n^(n+1)-(n-1)^n,","))

Edited and extended by Klaus Brockhaus, May 30 2003

A062815 a(n) = Sum_{i=1..n} i^(i+1).

Original entry on oeis.org

1, 9, 90, 1114, 16739, 296675, 6061476, 140279204, 3627063605, 103627063605, 3242055440326, 110235260819398, 4047611646518687, 159615707204330911, 6728024062917221536, 301875929242270047392, 14364960381309995038401

Offset: 1

Olivier Gérard, Jun 23 2001

Seiichi Manyama, Table of n, a(n) for n = 1..385

Cf. A226065, A226140.

Partial sums of A007778. - Michel Marcus, Mar 26 2019

Accumulate[#^(#+1)&/@Range[17]]  (* Harvey P. Dale, Mar 18 2011 *)

a(n) = sum(i=1, n, i^(i+1)); \\ Michel Marcus, Mar 26 2019

Definition simplified by Jon E. Schoenfield, Nov 29 2008

A134362 a(n) is the number of functions f:X->X, where |X| = n, such that for every x in X, f(f(x)) != x (i.e., the square of the function has no fixed points; note this implies that the function has no fixed points).

Original entry on oeis.org

1, 0, 0, 2, 30, 444, 7360, 138690, 2954364, 70469000, 1864204416, 54224221050, 1721080885480, 59217131089908, 2195990208122880, 87329597612123594, 3707783109757616400, 167411012044894728720, 8010372386879991018496, 404912918159552083622130

Offset: 0

Adam Day (adam.r.day(AT)gmail.com), Jan 17 2008

nonn

This sequence arose when analyzing the Zen Stare game. This game is played with a group of people standing in a circle. They start heads bowed and then everyone raises their heads simultaneously and looks at someone else in the circle. If no two people are looking at each other a Zen Stare is achieved.

			a(3) = 2 because given a three-element set X:= {A, B, C} the only functions whose square has no fixed points are f:X->X where f(A)=B, f(B)=C, f(C)=A and g:X->X where g(A)=C, g(B)=A, g(C)=B.

Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60. Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

Alois P. Heinz, Table of n, a(n) for n = 0..150
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, page 130.
Marko Riedel, Proof of EGF and closed form

Cf. A065440, A007778.

a:= n -> (n-1)^n + add((-1)^i*mul(binomial(n-2*(j-1),2),j=1..i)*(n-1)^(n-2*i)/i!,i=1..floor(n/2)): seq(a(n), n=0..20);

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[-t - t^2/2]/(1 - t), {x, 0, nn}], x], 1] (* Geoffrey Critzer, Feb 06 2012 *)

a(n) = n!*sum(q=0, n\2, ((-1)^q/(2^q*q!)*(n-1)^(n-2*q)/(n-2*q)!)); \\ Michel Marcus, Mar 09 2016

E.g.f.: exp(-T(x)-T(x)^2/2)/(1-T(x)) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 06 2012
a(n) ~ exp(-3/2) * n^n. - Vaclav Kotesovec, Aug 16 2013
a(n) = n!*Sum_{q=0..floor(n/2)} ((-1)^q/(2^q q!) * (n-1)^(n-2q)/(n-2q)!). - Marko Riedel, Mar 08 2016

A176043 a(n) = (2n-1)(n-1)^(n-1).

Original entry on oeis.org

1, 1, 3, 20, 189, 2304, 34375, 606528, 12353145, 285212672, 7360989291, 210000000000, 6562168424053, 222902511206400, 8177627877990831, 322248197941182464, 13574710601806640625, 608742554432415203328, 28953409166021786746195, 1455817098785971890290688, 77158366570752229975835181

Offset: 0

Michel Lagneau, Apr 07 2010

Determinant of the symmetric n X n matrix M_n where M_n(j,k) = n for j = k, M_n(j,k) = 1 otherwise.
The eigenvalues are 2*n-1, and n-1 with multiplicity n-1. The determinant of M_n is (2n-1)*(n-1)^(n-1), where 0^0 = 1.
Number of functions from [n] to [n] with zero or one fixed point. - Olivier Gérard, Jul 31 2016

			a(5) = determinant(M_5) = 2304 where M_5 is the matrix
  [5 1 1 1 1]
  [1 5 1 1 1]
  [1 1 5 1 1]
  [1 1 1 5 1]
  [1 1 1 1 5]
The 20 functions from [3] to [3] with one or zero fixed point are:
  0fp : 211,212,231,232,311,312,331,332
  1fp : 111,112,131,132,   221,222,321,322,   213,233,313,333

Vincenzo Librandi, Table of n, a(n) for n = 0..100, (corrected by _Peter Luschny_, Jan 19 2019)

Cf. A174964.
Cf. A007778 (functions from [n] to [n] without fixed point).
Cf. A055897 (functions from [n] to [n] with one fixed point).
Cf. A212291 (bijections of [n] with zero or one fixed point).
Cf. A000166 (bijections of [n] without fixed point).
Cf. A000240 (bijections of [n] with one fixed point).

[ (2*n-1)*(n-1)^(n-1): n in [1..17] ]; // Klaus Brockhaus, Apr 19 2010

[ Determinant( SymmetricMatrix( &cat[ [ j eq k select n else 1: k in [1..j] ]: j in [1..n] ] ) ): n in [1..17] ]; // Klaus Brockhaus, Apr 19 2010

for n from 2 to 30 do:x:=(2*n-1)*(n-1)^(n-1):print(x) :od:

Join[{1},Table[(2n-1)(n-1)^(n-1),{n,2,20}]] (* Harvey P. Dale, Jan 16 2014 *)

a(n)=n--; (2*n+1)*n^n \\ Charles R Greathouse IV, Jul 31 2016

a(n) = (2*n-1)*(n-1)^(n-1).
A176043(n) = A007778(n-1) + A055897(n).
a(n+1) = n! * [x^n] exp(n*x)*(1 + 2*n*x) for n >= 0. - Stefano Spezia, May 07 2023

Edited by Klaus Brockhaus, Apr 19 2010

New interpretation and cross-references by Olivier Gérard, Jul 31 2016

A350157 Total number of nodes in the smallest connected component summed over all endofunctions on [n].

Original entry on oeis.org

0, 1, 7, 61, 709, 9911, 167111, 3237921, 71850913, 1780353439, 49100614399, 1482061739423, 48873720208853, 1740252983702871, 66793644836081827, 2740470162691675711, 120029057782404141841, 5575505641199441262767, 274412698693082818767335, 14236421024010426118259883

Offset: 0

Alois P. Heinz, Dec 17 2021

nonn

			a(2) = 7 = 2 + 2 + 1 + 2: 11, 22, 12, 21.

Alois P. Heinz, Table of n, a(n) for n = 0..385

Column k=1 of A350202.

Cf. A000312, A001865, A007778, A209327, A347999.

g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(
      b(n-i, min(m, i))*g(i)*binomial(n-1, i-1), i=1..n))
    end:
a:= n-> (p-> add(coeff(p, x, i)*i, i=0..n))(b(n,n)):
seq(a(n), n=0..23);

g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[
     b[n - i, Min[m, i]]*g[i]*Binomial[n - 1, i - 1], {i, 1, n}]];
a[n_] := Function[p, Sum[Coefficient[p, x, i]*i, {i, 0, n}]][b[n, n]];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 27 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A347999(n,k).

cp's OEIS Frontend

A008787 a(n) = (n + 6)^n.

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A008790 a(n) = n^(n+4).

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A283498 a(n) = Sum_{d|n} d^(d+1).

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A085606 a(n) = (n-1)^n - 1.

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A110567 a(n) = n^(n+1) + 1.

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A084363 a(n) = n^(n+1) - (n-1)^n.

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A062815 a(n) = Sum_{i=1..n} i^(i+1).

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A176043 a(n) = (2n-1)(n-1)^(n-1).