A174962 - OEIS

%I #19 Sep 08 2022 08:45:51

%S 2,10,81,896,12500,209952,4117715,92274688,2324522934,65000000000,

%T 1997181694277,66870753361920,2423000852738024,94452058017243136,

%U 3941045013427734375,175244068700240740352,8272402618863367641770

%N a(n) = n^n*(3+n)/2.

%C Also determinant of the n X n matrix M_n with M_n(j,k) = j for j <> k, M_n(j,k) = n+j for j = k.

%C The eigenvalues of M_n are n+n(n+1)/2, and n with multiplicity n-1; cf. reference for proof. The determinant of M_n is n^n*(3+n)/2.

%D J.-M. Monier, Algèbre et géometrie, exercices corrigés. Dunod, 1997, p. 78.

%H Vincenzo Librandi, <a href="/A174962/b174962.txt">Table of n, a(n) for n = 1..100</a>

%e (in Maple notation) For n = 1, det(matrix(1,1,[[2]])) = 2; for n = 2, det(matrix(2,2,[[3,1],[2,4]])) = 10; for n = 3, det(matrix(3,3,[[4,1,1],[2,5,2],[3,3,6]])) = 81; for n = 4, det(matrix(4,4,[[5,1,1,1],[2,6,2,2],[3,3,7,3],[4,4,4,8]])) = 896.

%p for n from 1 to 25 do: x:=n^n *(3+n)/2:print(x):od:

%t Table[n^n(3+n)/2,{n,20}] (* _Harvey P. Dale_, May 04 2012 *)

%o (Magma) [ n^n*(3+n)/2: n in [1..17] ]; // _Klaus Brockhaus_, Apr 06 2010

%o (Magma) [ Determinant( Matrix([ &cat[ [j ne k select j else n+j]: k in [1..n] ]: j in [1..n] ]) ): n in [1..17] ]; // _Klaus Brockhaus_, Apr 06 2010

%K nonn

%O 1,1

%A _Michel Lagneau_, Apr 02 2010

%E Edited by _Klaus Brockhaus_, Apr 06 2010