

A176043


a(n) = (2*n1)*(n1)^(n1).


2



1, 1, 3, 20, 189, 2304, 34375, 606528, 12353145, 285212672, 7360989291, 210000000000, 6562168424053, 222902511206400, 8177627877990831, 322248197941182464, 13574710601806640625, 608742554432415203328
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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*n1, and n1 with multiplicity n1. The determinant of M_n is (2n1)*(n1)^(n1), 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*n1)*(n1)^(n1).
A176043(n) = A007778(n1) + A055897(n).


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*n1)*(n1)^(n1):print(x) :od:


MATHEMATICA

Join[{1}, Table[(2n1)(n1)^(n1), {n, 2, 20}]] (* Harvey P. Dale, Jan 16 2014 *)


PROG

(MAGMA) [ (2*n1)*(n1)^(n1): 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 A120485 A087152
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 crossreferences by Olivier Gérard, Jul 31 2016


STATUS

approved



