cp's OEIS Frontend

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

%I A176043 #30 May 08 2023 09:35:03
%S A176043 1,1,3,20,189,2304,34375,606528,12353145,285212672,7360989291,
%T A176043 210000000000,6562168424053,222902511206400,8177627877990831,
%U A176043 322248197941182464,13574710601806640625,608742554432415203328,28953409166021786746195,1455817098785971890290688,77158366570752229975835181
%N A176043 a(n) = (2*n-1)*(n-1)^(n-1).
%C A176043 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.
%C A176043 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.
%C A176043 Number of functions from [n] to [n] with zero or one fixed point. - _Olivier Gérard_, Jul 31 2016
%H A176043 Vincenzo Librandi, <a href="/A176043/b176043.txt">Table of n, a(n) for n = 0..100</a>, (corrected by _Peter Luschny_, Jan 19 2019)
%F A176043 a(n) = (2*n-1)*(n-1)^(n-1).
%F A176043 A176043(n) = A007778(n-1) + A055897(n).
%F A176043 a(n+1) = n! * [x^n] exp(n*x)*(1 + 2*n*x) for n >= 0. - _Stefano Spezia_, May 07 2023
%e A176043 a(5) = determinant(M_5) = 2304 where M_5 is the matrix
%e A176043   [5 1 1 1 1]
%e A176043   [1 5 1 1 1]
%e A176043   [1 1 5 1 1]
%e A176043   [1 1 1 5 1]
%e A176043   [1 1 1 1 5]
%e A176043 The 20 functions from [3] to [3] with one or zero fixed point are:
%e A176043   0fp : 211,212,231,232,311,312,331,332
%e A176043   1fp : 111,112,131,132,   221,222,321,322,   213,233,313,333
%p A176043 for n from 2 to 30 do:x:=(2*n-1)*(n-1)^(n-1):print(x) :od:
%t A176043 Join[{1},Table[(2n-1)(n-1)^(n-1),{n,2,20}]] (* _Harvey P. Dale_, Jan 16 2014 *)
%o A176043 (Magma) [ (2*n-1)*(n-1)^(n-1): n in [1..17] ]; // _Klaus Brockhaus_, Apr 19 2010
%o A176043 (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
%o A176043 (PARI) a(n)=n--; (2*n+1)*n^n \\ _Charles R Greathouse IV_, Jul 31 2016
%Y A176043 Cf. A174964.
%Y A176043 Cf. A007778 (functions from [n] to [n] without fixed point).
%Y A176043 Cf. A055897 (functions from [n] to [n] with one fixed point).
%Y A176043 Cf. A212291 (bijections of [n] with zero or one fixed point).
%Y A176043 Cf. A000166 (bijections of [n] without fixed point).
%Y A176043 Cf. A000240 (bijections of [n] with one fixed point).
%K A176043 nonn,easy
%O A176043 0,3
%A A176043 _Michel Lagneau_, Apr 07 2010
%E A176043 Edited by _Klaus Brockhaus_, Apr 19 2010
%E A176043 New interpretation and cross-references by _Olivier Gérard_, Jul 31 2016