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A118714 Determinant of n X n matrix whose diagonal contains the first n tetrahedral numbers and all other elements are 1's.

%I A118714 #16 Apr 17 2018 05:02:32
%S A118714 1,3,27,513,17442,959310,79622730,9475104870,1553917198680,
%T A118714 340307866510920,96987741955612200,35206550329887228600,
%U A118714 15983773849768801784400,8934929582020760197479600,6066817186192096174088648400,4944456006746558381882248446000
%N A118714 Determinant of n X n matrix whose diagonal contains the first n tetrahedral numbers and all other elements are 1's.
%C A118714 a(n+2) / a(n+1) = A062748(n) = A062745(n+2, 3)= binomial(n+4, 3)-1 = (n+1)*(n^2+8*n+18)/3!.
%H A118714 Colin Barker, <a href="/A118714/b118714.txt">Table of n, a(n) for n = 1..150</a>
%F A118714 a(n) = Det[ DiagonalMatrix[ Table[ i*(i+1)(i+2)/6 - 1, {i, 1, n} ] ] + 1 ].
%F A118714 a(n) = Product[(j-3)*(j^2+2)/3!,{j,4,n+2}].
%F A118714 a(n) = Product[(k+1)*(k^2+8*k+18)/3!,{k,0,n-2}] = Product[A062748(k),{k,0,n-2}].
%F A118714 a(n) ~ sqrt(Pi) * sinh(Pi*sqrt(2)) * n^(3*n + 9/2) / (11 * 2^(n-1) * 3^(n+1) * exp(3*n)). - _Vaclav Kotesovec_, Apr 17 2018
%e A118714 The matrix begins:
%e A118714 1 1 1 1 1 1 1 ...
%e A118714 1 4 1 1 1 1 1 ...
%e A118714 1 1 10 1 1 1 1 ...
%e A118714 1 1 1 20 1 1 1 ...
%e A118714 1 1 1 1 35 1 1 ...
%e A118714 1 1 1 1 1 56 1 ...
%p A118714 a:= proc(n) option remember; `if`(n<2, 1,
%p A118714       a(n-1) *(6+4*n+n^2)*(n-1)/6)
%p A118714     end:
%p A118714 seq(a(n), n=1..20);  # _Alois P. Heinz_, Nov 15 2015
%t A118714 Table[ Det[ DiagonalMatrix[ Table[ i*(i+1)(i+2)/6 - 1, {i, 1, n} ] ] + 1 ], {n, 1, 20} ]
%t A118714 Table[Product[(k-3)*(k^2+2)/3!,{k,4,n+2}],{n,1,20}]
%o A118714 (PARI) a(n) = matdet(matrix(n, n, i, j, if(i==j, i*(i+1)*(i+2)/6, 1))) \\ _Colin Barker_, Nov 13 2015
%Y A118714 Cf. A000292, A067550, A062748, A062745.
%K A118714 nonn
%O A118714 1,2
%A A118714 _Alexander Adamchuk_, May 20 2006
%E A118714 a(15) and a(16) from _Colin Barker_, Nov 13 2015