A062029 - OEIS

%I #16 May 09 2022 16:16:53

%S 2,24,960,80640,11531520,2500485120,763847884800,312344808652800,

%T 164644289755545600,108684799028822016000,87805845811395506995200,

%U 85211145316323008446464000,97803969545162680178835456000

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

%H Harry J. Smith, <a href="/A062029/b062029.txt">Table of n, a(n) for n=1..100</a>

%F a(n) = Product_{k=1..n} (n^2 - n + 2*k) = (n^2 + n)!!/(n^2 - n)!! .

%F a(n) = 2^n*Gamma((n^2 + n + 2)/2)/Gamma(n^2 - n + 2)/2).

%F a(n) = 2^n * A057003(n-1).

%e a(3) = 8*10*12 = 960.

%t Table[2^n*Gamma[(2+n+n^2)/2]/Gamma[(2-n+n^2)/2], {n,30}] (* _G. C. Greubel_, May 05 2022 *)

%t With[{nn=30},Times@@@TakeList[Range[2,(nn(nn+1))/2,2],Range[nn/2]]] (* _Harvey P. Dale_, May 09 2022 *)

%o (PARI) { for (n=1, 60, write("b062029.txt", n, " ", 2^n*((n^2 + n)/2)!/((n^2 - n)/2)!) ) } \\ _Harry J. Smith_, Jul 30 2009

%o (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 Cf. A057003, A062030, A062031, A062032.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Jun 02 2001

%E Formula and more terms from _Vladeta Jovovic_, Jun 05 2001