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Group odd numbers into (1), (3,5,7), (9,11,13,15,17), ...; a(n) = product of n-th group.
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%I #13 May 06 2022 19:55:54

%S 1,105,328185,5568833025,304513870485825,40992233865440682825,

%T 11492457771692770753505625,5984524775454356180393209490625,

%U 5325142910343897163530366857379506625,7598549164899334249502031499667984969915625

%N Group odd numbers into (1), (3,5,7), (9,11,13,15,17), ...; a(n) = product of n-th group.

%H Harry J. Smith, <a href="/A062031/b062031.txt">Table of n, a(n) for n=1..100</a>

%F a(n) = Product_{k=0..2*n-2} (2*k + 2*n*(n-2) + 3). - _Harry J. Smith_, Jul 30 2009

%F a(n) = (Gamma(2*n^2 + 1)*Gamma((n-1)^2 + 1))/(2^(2*n-1)*Gamma(n^2 + 1)*Gamma(2*(n-1)^2 + 1)). - _G. C. Greubel_, May 06 2022

%e a(2) = 3*5*7 = 105.

%t Table[(Gamma[2*n^2 +1]*Gamma[(n-1)^2 +1])/(2^(2*n-1)*Gamma[n^2 +1]*Gamma[2*(n-1)^2 +1]), {n, 30}] (* _G. C. Greubel_, May 06 2022 *)

%o (PARI) { for (n=1, 100, b=2*n^2 - 4*n + 3; write("b062031.txt", n, " ", prod(k=0, 2*n - 2, b + 2*k)) ) } \\ _Harry J. Smith_, Jul 30 2009

%o (SageMath) [(gamma(2*n^2 +1)*gamma((n-1)^2 +1))/(2^(2*n-1)*gamma(n^2 +1)*gamma(2*(n-1)^2 +1)) for n in (1..30)] # _G. C. Greubel_, May 06 2022

%Y Cf. A062029, A062030, A062032.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Jun 02 2001

%E More terms from _Matthew Conroy_, Jun 11 2001