A174962 a(n) = n^n*(3+n)/2.

2, 10, 81, 896, 12500, 209952, 4117715, 92274688, 2324522934, 65000000000, 1997181694277, 66870753361920, 2423000852738024, 94452058017243136, 3941045013427734375, 175244068700240740352, 8272402618863367641770

Offset: 1

Author

Michel Lagneau, Apr 02 2010

Keywords

• nonn

Comments

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.

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.

Examples

			(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.
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100

Programs

Magma

[ n^n*(3+n)/2: n in [1..17] ]; // Klaus Brockhaus, Apr 06 2010

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

Maple

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

Mathematica

Table[n^n(3+n)/2,{n,20}] (* Harvey P. Dale, May 04 2012 *)

Extensions

Edited by Klaus Brockhaus, Apr 06 2010