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).
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).
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