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A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

%I A107254 #45 Feb 16 2025 08:32:57
%S A107254 1,1,12,8640,870912000,22122558259200000,222531556847250309120000000,
%T A107254 1280394777025250130271722799104000000000,
%U A107254 5746332926632566442385615219551212618645504000000000000
%N A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.
%C A107254 Inverse product of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - _Alexander Adamchuk_, Apr 12 2006
%C A107254 The n X n matrix with A(i,j) = 1/(i+j-1)! (i,j = 1..n) has determinant (-1)^floor(n/2)/a(n). - _Mikhail Lavrov_, Nov 01 2022
%H A107254 Alois P. Heinz, <a href="/A107254/b107254.txt">Table of n, a(n) for n = 0..20</a>
%H A107254 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4566780">Determinant of a matrix involving factorials</a>.
%H A107254 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.
%F A107254 a(n) = n!*(n+1)!*(n+2)!*...*(2n-1)!/(0!*1!*2!*3!*...*(n-1)!) = A000178(2n-1)/A000178(n-1)^2 = A079478(n)/A000984(n) = A079478(n-1)*A009445(n-1) = A107252(n)*A000142(n) = A088020(n)/A039622(n).
%F A107254 a(n) = 1/Product_{j=1..n} ( Product_{i=1..n} 1/(i+j-1) ). - _Alexander Adamchuk_, Apr 12 2006
%F A107254 a(n) = 2^(n*(n-1)) * A136411(n) for n > 0 . - _Robert Coquereaux_, Apr 06 2013
%F A107254 a(n) = A136411(n) * A053763(n) for n > 0. [Following remark from _Robert Coquereaux_] - _M. F. Hasler_, Apr 06 2013
%F A107254 a(n) ~ A * 2^(2*n^2-1/12) * n^(n^2+1/12) / exp(3*n^2/2+1/12), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - _Vaclav Kotesovec_, Feb 10 2015
%F A107254 a(n) = Product_{k=1..n} rf(k,n) where rf denotes the rising factorial. - _Peter Luschny_, Nov 29 2015
%F A107254 a(n) = (n! * G(2*n+1))/(G(n+1)*G(n+2)), where G(n) is the Barnes G - function. - _G. C. Greubel_, Apr 21 2021
%e A107254 a(3) = 1!*2!*3!*4!*5!/(1!*2!*1!*2!) = 34560/4 = 8640.
%e A107254 n = 2: HilbertMatrix[n,n]
%e A107254   1/1 1/2
%e A107254   1/2 1/3
%e A107254 so a(2) = 1 / (1 * 1/2 * 1/2 * 1/3) = 12.
%e A107254 The n X n Hilbert matrix begins:
%e A107254   1/1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
%e A107254   1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
%e A107254   1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
%e A107254   1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
%e A107254   1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
%e A107254   1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
%p A107254 a:= n-> mul((n+i)!/i!, i=0..n-1):
%p A107254 seq(a(n), n=0..10);  # _Alois P. Heinz_, Jul 23 2012
%t A107254 Table[Product[(i+j-1),{i,1,n},{j,1,n}], {n,1,10}] (* _Alexander Adamchuk_, Apr 12 2006 *)
%t A107254 Table[n!*BarnesG[2n+1]/(BarnesG[n+2]*BarnesG[n+1]), {n,0,12}] (* _G. C. Greubel_, Apr 21 2021 *)
%o A107254 (Sage)
%o A107254 a = lambda n: prod(rising_factorial(k,n) for k in (1..n))
%o A107254 print([a(n) for n in (0..10)]) # _Peter Luschny_, Nov 29 2015
%o A107254 (Magma)
%o A107254 A107254:= func< n | n eq 0 select 1 else (&*[Factorial(n+j)/Factorial(j): j in [0..n-1]]) >;
%o A107254 [A107254(n): n in [0..12]]; // _G. C. Greubel_, Apr 21 2021
%Y A107254 Cf. A000178, A002457, A005249, A098118.
%K A107254 nonn
%O A107254 0,3
%A A107254 _Henry Bottomley_, May 14 2005