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A062029 Group even numbers into (2), (4,6), (8,10,12), (14,16,18,20), ...; a(n) = product of n-th group.

%I A062029 #24 Jan 07 2025 05:56:59
%S A062029 2,24,960,80640,11531520,2500485120,763847884800,312344808652800,
%T A062029 164644289755545600,108684799028822016000,87805845811395506995200,
%U A062029 85211145316323008446464000,97803969545162680178835456000,131047222390590123375392194560000,202702319752278628965061257854976000
%N A062029 Group even numbers into (2), (4,6), (8,10,12), (14,16,18,20), ...; a(n) = product of n-th group.
%H A062029 Harry J. Smith, <a href="/A062029/b062029.txt">Table of n, a(n) for n=1..100</a>
%F A062029 a(n) = Product_{k=1..n} (n^2 - n + 2*k) = (n^2 + n)!!/(n^2 - n)!! .
%F A062029 a(n) = 2^n*Gamma((n^2 + n + 2)/2)/Gamma((n^2 - n + 2)/2).
%F A062029 a(n) = 2^n * A057003(n-1).
%e A062029 a(3) = 8*10*12 = 960.
%t A062029 Table[2^n*Gamma[(2+n+n^2)/2]/Gamma[(2-n+n^2)/2], {n,30}] (* _G. C. Greubel_, May 05 2022 *)
%t A062029 With[{nn=30},Times@@@TakeList[Range[2,(nn(nn+1))/2,2],Range[nn/2]]] (* _Harvey P. Dale_, May 09 2022 *)
%o A062029 (PARI) a(n) = { 2^n*((n^2 + n)/2)!/((n^2 - n)/2)! } \\ _Harry J. Smith_, Jul 30 2009
%o A062029 (SageMath) [2^n*gamma((2+n+n^2)/2)/gamma((2-n+n^2)/2) for n in (1..30)] # _G. C. Greubel_, May 05 2022
%Y A062029 Cf. A057003, A062030, A062031, A062032.
%K A062029 nonn
%O A062029 1,1
%A A062029 _Amarnath Murthy_, Jun 02 2001
%E A062029 Formula and more terms from _Vladeta Jovovic_, Jun 05 2001