A055869 a(n) = (n+1)^n - n^n.

1, 5, 37, 369, 4651, 70993, 1273609, 26269505, 612579511, 15937424601, 457696700077, 14381984674225, 490839666661891, 18080919199832609, 715027614225987601, 30214447801957316865, 1358671297852359767791, 64780942222614703957417, 3264460344339686410876021

Offset: 1

Author

Wolfdieter Lang, Jun 20 2000

Keywords

• nonn
• easy

Comments

Number of functions f:[n]->[n+1] such that some x in [n] maps to n+1.

Number of switching generators for a power polyadic n-context ({1..k}, ..., {1..k}, <>) with n=k [Theorems 5 and 6, page 81, in Ignatov]. - Dmitry I. Ignatov, Nov 23 2022

Links

• G. C. Greubel, Table of n, a(n) for n = 1..385
• D. I. Ignatov, On closure operators related to maximal tricliques in tripartite hypergraphs, Discrete Appl. Math., 249 (2018), 74-84.
• D. I. Ignatov, Supporting iPython code for enumeration of switching generators of power polyadic n-contexts for n=k<6, GitHub repository.

Crossrefs

Row sums of triangle A055864.

Cf. A055864, A055858, A045531.

Programs

Magma

[(n+1)^n - n^n: n in [1..40]]; // Vincenzo Librandi, Jan 11 2015

Mathematica

Table[(n+1)^n-n^n,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009 *)

PARI

vector(20, n, (n+1)^n - n^n) \\ Michel Marcus, Jan 10 2015

Formula

E.g.f.: W(-x)*(x-1)/((1+W(-x))*x), W(x) principal branch of Lambert's function.

a(n) = Sum_{m=1..n} A055864(n, m).

a(n) = Sum_{i=0..n-1} n^i*C(n, i). - Olivier Gérard, Jun 26 2001

With interpolated zeros, ceiling(n/2)^floor(n/2) - floor(n/2)^floor(n/2). - Paul Barry, Jul 13 2005

a(n) = Sum_{k=1..n} (-1)^(n-k)*k!*Stirling2(n,k)*binomial(n+k-1,n). - Vladimir Kruchinin, Sep 20 2015

Extensions

More terms from Vincenzo Librandi, Jan 11 2015