A062030
Group even numbers into (2,4), (6,8,10,12), (14,16,18,20,22,24), ...; a(n) = product of n-th group.
Original entry on oeis.org
8, 5760, 42577920, 1300252262400, 111644006842368000, 21695920874860629196800, 8291067715225260172247040000, 5644260808699395278689265516544000, 6360332664265371581768550654463180800000, 11209384544297234954537967755979151481241600000, 29531169256166572959626706182319305835700813824000000
Offset: 1
a(3) = 14*16*18*20*22*24 = 42577920.
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Table[4^n*Gamma[1+n+n^2]/Gamma[1-n+n^2], {n,30}] (* G. C. Greubel, May 05 2022 *)
Module[{nn=20,ev,l},ev=2*Range[nn(nn+1)];l=2*Range[nn];Times@@@TakeList[ev,l]] (* Harvey P. Dale, Nov 24 2024 *)
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a(n) = { 2^(2*n)*(n^2+n)!/(n^2-n)! } \\ Harry J. Smith, Jul 30 2009
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[4^n*gamma(1+n+n^2)/gamma(1-n+n^2) for n in (1..30)] # G. C. Greubel, May 05 2022
A062031
Group odd numbers into (1), (3,5,7), (9,11,13,15,17), ...; a(n) = product of n-th group.
Original entry on oeis.org
1, 105, 328185, 5568833025, 304513870485825, 40992233865440682825, 11492457771692770753505625, 5984524775454356180393209490625, 5325142910343897163530366857379506625, 7598549164899334249502031499667984969915625
Offset: 1
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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 *)
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a(n) = { my(b=2*n^2 - 4*n + 3); prod(k=0, 2*n - 2, b + 2*k) } \\ Harry J. Smith, Jul 30 2009
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[(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
A062032
Group odd numbers into (1), (3,5), (7,9,11), (13,15,17,19), ...; a(n) = product of n-th group.
Original entry on oeis.org
1, 15, 693, 62985, 9454725, 2118331215, 662496582825, 275735605996305, 147364622598587625, 98358760729571316975, 80185770642041047108125, 78405694972326706112753625, 90569612902695107431619494125, 122020670469540010360975931523375, 189638875693941730653122520269900625
Offset: 1
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len=20; Times@@@FoldPairList[TakeDrop,Range[1,len^2+len-1,2],Range[len]] (* The program uses the FoldPairList and TakeDrop functions from Mathematica version 10 *) (* Harvey P. Dale, Jul 29 2015 *)
Table[(Gamma[2*Binomial[n+1,2] +1]*Gamma[Binomial[n,2] +1])/(2^n*Gamma[Binomial[n +1,2] +1]*Gamma[2*Binomial[n,2] +1]), {n, 30}] (* G. C. Greubel, May 06 2022 *)
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{ for (n=1, 100, b=n^2 - n + 1; write("b062032.txt", n, " ", prod(k=0, n - 1, b + 2*k)) ) } \\ Harry J. Smith, Jul 30 2009
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[(gamma(2*binomial(n+1,2) +1)*gamma(binomial(n,2) +1))/(2^n*gamma(binomial(n+1,2) +1)*gamma(2*binomial(n,2) +1)) for n in (1..30)] # G. C. Greubel, May 06 2022
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