A061250 -id:A061250 - cp's OEIS frontend

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

0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632

Offset: 0

N. J. A. Sloane

Number of edges of the complete bipartite graph of order n+n^n, K_n,n^n. - Roberto E. Martinez II, Jan 07 2002
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n >= 1. - Nick Hobson, Nov 30 2006
a(n) is also the number of ways of writing an n-cycle as the product of n+1 transpositions. - Nikos Apostolakis, Nov 22 2008
a(n) is the total number of elements whose preimage is the empty set summed over all partial functions from [n] into [n]. - Geoffrey Critzer, Jan 12 2022

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 67.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Nick Hobson, Exponential equation.
Yidong Sun and Jujuan Zhuang, lambda-factorials of n, arXiv:1007.1339 [math.CO], 2010. - _Peter Luschny_, Jul 09 2010

Cf. A000169, A000272, A000312, A007830, A008785, A008786, A008787, A008788, A008789, A008790, A008791, A135608.
Essentially the same as A065440.
Cf. A061250, A143857. [From Reinhard Zumkeller, Jul 23 2010]

[n^(n+1):n in [0..20]]; // Vincenzo Librandi, Jan 03 2012

seq( n^(n+1), n=0..20); # G. C. Greubel, Mar 05 2020

Table[n^(n+1), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)

A007778[n]:=n^(n+1)$
makelist(A007778[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

vector(21, n, my(m=n-1); m^(m+1)) \\ G. C. Greubel, Mar 05 2020

[n^(n+1) for n in (0..20)] # G. C. Greubel, Mar 05 2020

E.g.f.: -W(-x)/(1 + W(-x))^3, W(x) Lambert's function (principal branch).
a(n) = Sum_{k=0..n} binomial(n,k)*A000166(k+1)*(n+1)^(n-k). - Peter Luschny, Jul 09 2010
See A008517 and A134991 for similar e.g.f.s. and A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d/dx {x/(T(x)*(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
a(n) = n*A000312(n). - R. J. Mathar, Jan 12 2017
Sum_{n>=2} 1/a(n) = A135608. - Amiram Eldar, Nov 17 2020

A143857 a(n) = n + (n+1)*(n+2)^(n+3).

Original entry on oeis.org

8, 163, 3074, 62503, 1399684, 34588811, 939524102, 27894275215, 900000000008, 31384283767219, 1176925259169802, 47248516628391479, 2022385242251558924, 91957716979980468763, 4427218577690292387854, 225009351233083599856159

Offset: 0

Reinhard Zumkeller, Sep 03 2008

Suggested by Karl Vago's contributions to Dario Alpern's list of records.

			a(1) = 1+(1+1)*(1+2)^(1+3) = 1+2*3^4 = 163.

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Dario A. Alpern, ECM Factorization applet records
Dario A. Alpern, Factorization using the Elliptic Curve Method

Cf. A007778, A061250.

List([0..20], n-> n + (n+1)*(n+2)^(n+3)); # G. C. Greubel, Nov 09 2019

[n+(n+1)*(n+2)^(n+3): n in [0..20]]; // Vincenzo Librandi, Dec 27 2010

A143857:=n->n+(n+1)*(n+2)^(n+3); seq(A143857(n), n=0..20); # Wesley Ivan Hurt, Mar 20 2014

Table[n + (n+1)*(n+2)^(n+3), {n, 0, 20}] (* Vincenzo Librandi, Mar 20 2014 *)

vector(21, n, (n-1) + n*(n+1)^(n+2)) \\ G. C. Greubel, Nov 09 2019

[n + (n+1)*(n+2)^(n+3) for n in (0..20)] # G. C. Greubel, Nov 09 2019

a(n) = n + A061250(n+3). - R. J. Mathar, Sep 04 2008

A061302 a(n) = n! * [x^n] W(-x)*(W(-x) + 2)/(W(-x) + 1), where W denotes Lambert's W function.

Original entry on oeis.org

0, 2, 6, 36, 320, 3750, 54432, 941192, 18874368, 430467210, 11000000000, 311249095212, 9659108818944, 326173191714734, 11905721598812160, 467086816406250000, 19599665578316398592, 875901453762003632658

Offset: 0

Gero Burghardt (gerogoestohollywood(AT)yahoo.de), Jun 05 2001

			2*x + 6*x^2 +36*x^3 + 320*x^4 + 3750*x^5 + 54432*x^6 + 941192*x^7 + ...

Stephan Wolfram, The Mathematica Book, 4th Edition, Cambridge University Press, section 3.2.10 'Special Functions', page 772, 1999.

Cf. A061250.

Essentially the same as A055541.

W := LambertW: egf := -W(-x)*(W(-x) + 2)/(W(-x) + 1):
ser := series(egf, x, 20): seq(n!*coeff(ser, x, n), n = 0..17); # Peter Luschny, Feb 10 2023

Range[18]!CoefficientList[ Series[ -ProductLog[ -x], {x, 0, 17}], x] (* Robert G. Wilson v, Mar 23 2005 *)
a[ n_] := If[ n < 0, 0, (n + 1)! SeriesCoefficient[ -ProductLog[-x], {x, 0, n}]] (* Michael Somos, Jun 07 2012 *)

a(n) = (n+1)*n^(n-1) with a(0) = 0.

Corrected and extended by Jason Earls, Jun 09 2001

Name made consistent with offset by Peter Luschny, Feb 10 2023

A344744 a(n) is the n-th power of the concatenation of the integers from 0 through n-1.

Original entry on oeis.org

0, 1, 1728, 228886641, 2861381721051424, 3539537889086624823140625, 437104634676747795452235896466702336, 5396563761318393964062660689603780554533710504641, 6662458388479360230805308787387369820914640828074410829911019008

Offset: 1

Luke Voyles, May 27 2021

			a(1) = 0^1 = 0;
a(2) = 01^2 = 1;
a(3) = 012^3 = 1728;
a(4) = 0123^4 = 228886641.

Cf. A007778, A007908, A061250.

a[n_] := FromDigits[Join @@ IntegerDigits @ Range[0, n - 1]]^n; Array[a, 9] (* Amiram Eldar, May 29 2021 *)

def a(n): return int("".join(str(i) for i in range(n)))**n
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, May 29 2021

a(n) = A007908(n-1)^n.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A143857 a(n) = n + (n+1)*(n+2)^(n+3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A061302 a(n) = n! * [x^n] W(-x)*(W(-x) + 2)/(W(-x) + 1), where W denotes Lambert's W function.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A344744 a(n) is the n-th power of the concatenation of the integers from 0 through n-1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula