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

%I A174962 #19 Sep 08 2022 08:45:51
%S A174962 2,10,81,896,12500,209952,4117715,92274688,2324522934,65000000000,
%T A174962 1997181694277,66870753361920,2423000852738024,94452058017243136,
%U A174962 3941045013427734375,175244068700240740352,8272402618863367641770
%N A174962 a(n) = n^n*(3+n)/2.
%C A174962 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 A174962 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 A174962 J.-M. Monier, Algèbre et géometrie, exercices corrigés. Dunod, 1997, p. 78.
%H A174962 Vincenzo Librandi, <a href="/A174962/b174962.txt">Table of n, a(n) for n = 1..100</a>
%e A174962 (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 A174962 for n from 1 to 25 do: x:=n^n *(3+n)/2:print(x):od:
%t A174962 Table[n^n(3+n)/2,{n,20}] (* _Harvey P. Dale_, May 04 2012 *)
%o A174962 (Magma) [ n^n*(3+n)/2: n in [1..17] ]; // _Klaus Brockhaus_, Apr 06 2010
%o A174962 (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 A174962 nonn
%O A174962 1,1
%A A174962 _Michel Lagneau_, Apr 02 2010
%E A174962 Edited by _Klaus Brockhaus_, Apr 06 2010