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a(n) = (n-1)!(n+2)!/(3*2^n).
12

%I #19 Feb 02 2023 02:29:52

%S 1,2,10,90,1260,25200,680400,23814000,1047816000,56582064000,

%T 3677834160000,283193230320000,25487390728800000,2650688635795200000,

%U 315431947659628800000,42583312934049888000000,6472663565975582976000000,1100352806215849105920000000

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

%C Determinant of n X n matrix whose diagonal are the first n triangular numbers and all other elements are 1's.

%H Muniru A Asiru, <a href="/A067550/b067550.txt">Table of n, a(n) for n = 1..120</a>

%F a(n+1)/a(n) = A000096(n) = n(n+3)/2. - _Alexander Adamchuk_, May 20 2006

%F From _Amiram Eldar_, Feb 02 2023: (Start)

%F Sum_{n>=1} 1/a(n) = 3*BesselI(3, 2*sqrt(2))/sqrt(2).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*BesselJ(3, 2*sqrt(2))/sqrt(2). (End)

%e The determinant begins:

%e 1 1 1 1 1 1 1 ...

%e 1 3 1 1 1 1 1 ...

%e 1 1 6 1 1 1 1 ...

%e 1 1 1 10 1 1 1 ...

%e 1 1 1 1 15 1 1 ...

%e 1 1 1 1 1 21 1 ...

%p d:=(i,j)->`if`(i<>j,1,i*(i+1)/2):

%p seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20); # _Muniru A Asiru_, Mar 05 2018

%t Table[ Det[ DiagonalMatrix[ Table[ i(i + 1)/2 - 1, {i, 1, n} ] ] + 1 ], {n, 1, 20} ]

%t Table[(n-1)!(n+2)!/3/2^n,{n,1,20}] (* _Alexander Adamchuk_, May 20 2006 *)

%o (GAP) A067550:=List([1..20],n->Factorial(n-1)*Factorial(n+2)/(3*2^n)); # _Muniru A Asiru_, Mar 05 2018

%o (PARI) a(n) = (n-1)!*(n+2)!/(3*2^n); \\ _Altug Alkan_, Mar 05 2018

%Y Cf. A000096.

%K nonn

%O 1,2

%A _Robert G. Wilson v_, Jan 28 2002

%E a(18) from _Muniru A Asiru_, Mar 05 2018