OFFSET
1,1
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.
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
EXAMPLE
(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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Apr 02 2010
EXTENSIONS
Edited by Klaus Brockhaus, Apr 06 2010
STATUS
approved