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A067550 a(n) = (n-1)!(n+2)!/(3*2^n).

%I A067550 #19 Feb 02 2023 02:29:52
%S A067550 1,2,10,90,1260,25200,680400,23814000,1047816000,56582064000,
%T A067550 3677834160000,283193230320000,25487390728800000,2650688635795200000,
%U A067550 315431947659628800000,42583312934049888000000,6472663565975582976000000,1100352806215849105920000000
%N A067550 a(n) = (n-1)!(n+2)!/(3*2^n).
%C A067550 Determinant of n X n matrix whose diagonal are the first n triangular numbers and all other elements are 1's.
%H A067550 Muniru A Asiru, <a href="/A067550/b067550.txt">Table of n, a(n) for n = 1..120</a>
%F A067550 a(n+1)/a(n) = A000096(n) = n(n+3)/2. - _Alexander Adamchuk_, May 20 2006
%F A067550 From _Amiram Eldar_, Feb 02 2023: (Start)
%F A067550 Sum_{n>=1} 1/a(n) = 3*BesselI(3, 2*sqrt(2))/sqrt(2).
%F A067550 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*BesselJ(3, 2*sqrt(2))/sqrt(2). (End)
%e A067550 The determinant begins:
%e A067550   1  1  1  1  1  1  1 ...
%e A067550   1  3  1  1  1  1  1 ...
%e A067550   1  1  6  1  1  1  1 ...
%e A067550   1  1  1 10  1  1  1 ...
%e A067550   1  1  1  1 15  1  1 ...
%e A067550   1  1  1  1  1 21  1 ...
%p A067550 d:=(i,j)->`if`(i<>j,1,i*(i+1)/2):
%p A067550 seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20); # _Muniru A Asiru_, Mar 05 2018
%t A067550 Table[ Det[ DiagonalMatrix[ Table[ i(i + 1)/2 - 1, {i, 1, n} ] ] + 1 ], {n, 1, 20} ]
%t A067550 Table[(n-1)!(n+2)!/3/2^n,{n,1,20}] (* _Alexander Adamchuk_, May 20 2006 *)
%o A067550 (GAP) A067550:=List([1..20],n->Factorial(n-1)*Factorial(n+2)/(3*2^n)); # _Muniru A Asiru_, Mar 05 2018
%o A067550 (PARI) a(n) = (n-1)!*(n+2)!/(3*2^n); \\ _Altug Alkan_, Mar 05 2018
%Y A067550 Cf. A000096.
%K A067550 nonn
%O A067550 1,2
%A A067550 _Robert G. Wilson v_, Jan 28 2002
%E A067550 a(18) from _Muniru A Asiru_, Mar 05 2018