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
A062029
Group even numbers into (2), (4,6), (8,10,12), (14,16,18,20), ...; a(n) = product of n-th group.
Original entry on oeis.org
2, 24, 960, 80640, 11531520, 2500485120, 763847884800, 312344808652800, 164644289755545600, 108684799028822016000, 87805845811395506995200, 85211145316323008446464000, 97803969545162680178835456000, 131047222390590123375392194560000, 202702319752278628965061257854976000
Offset: 1
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Table[2^n*Gamma[(2+n+n^2)/2]/Gamma[(2-n+n^2)/2], {n,30}] (* G. C. Greubel, May 05 2022 *)
With[{nn=30},Times@@@TakeList[Range[2,(nn(nn+1))/2,2],Range[nn/2]]] (* Harvey P. Dale, May 09 2022 *)
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a(n) = { 2^n*((n^2 + n)/2)!/((n^2 - n)/2)! } \\ Harry J. Smith, Jul 30 2009
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[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
A062079
Group the odd numbers as (1), (3,5), (7,9,11), (13,15,17,19), (21,23,25,27,29), ... then a(n) = LCM of the n-th group.
Original entry on oeis.org
1, 15, 693, 62985, 3151575, 706110405, 44166438855, 30637289555145, 3274769391079725, 312250034062131165, 593968671422526274875, 5531265959247033940935, 95840860214492177176316925
Offset: 1
a(3) = lcm(7,9,11) = 693.
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Table[LCM[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,20}] (* G. C. Greubel, May 13 2022 *)
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a(n) = local(r);r=1;forstep(k=n^2-n+1,n^2+n-1,2,r=lcm(r,k));r \\ Franklin T. Adams-Watters, Jul 03 2009
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{ for (n=1, 100, a=b=n^2 - n + 1; for (k=1, n - 1, a=lcm(a, b + 2*k)); write("b062079.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 31 2009
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[lcm(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..20)] # G. C. Greubel, May 13 2022
Showing 1-4 of 4 results.
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