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

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

Offset

0,3

Comments

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

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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

Prog

(Magma) [ (2*n-1)*(n-1)^(n-1): n in [1..17] ]; // Klaus Brockhaus, Apr 19 2010
(Magma) [ 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
(PARI) a(n)=n--; (2*n+1)*n^n \\ Charles R Greathouse IV, Jul 31 2016

Crossrefs

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).

Sequence in context: A226349 A208975 A286794 * A108206 A349959 A120485

Adjacent sequences: A176040 A176041 A176042 * A176044 A176045 A176046

Keyword

nonn,easy

Author

Michel Lagneau, Apr 07 2010

Extensions

Edited by Klaus Brockhaus, Apr 19 2010

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

Status

approved