A263430 a(n) = Product_{k=0..n} (4*k+1)^(n-k).

1, 1, 5, 225, 131625, 1309010625, 273380323978125, 1427352844030287890625, 216119240915841469025244140625, 1079864992142473709995957417730712890625, 199639840782299404795675492100337942688751220703125

Offset: 0

Author

Vaclav Kotesovec, Oct 18 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..36
• Eric Weisstein's World of Mathematics, Catalan's Constant
• Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant

Crossrefs

Cf. A000178, A057863, A263416.

Programs

Magma

[(&*[(4*k+1)^(n-k): k in [0..n]]): n in [0..10]]; // G. C. Greubel, Aug 25 2018

Mathematica

Table[Product[(4*k+1)^(n-k),{k,0,n}],{n,0,10}]

PARI

for(n=0,10, print1(prod(k=0,n, (4*k+1)^(n-k)), ", ")) \\ G. C. Greubel, Aug 25 2018

Formula

a(n) ~ A^(1/8) * 2^(n^2 + 3*n/2 + 1/8) * Pi^(n/2 + 1/8) * n^(n^2/2 + n/4 - 5/96) / (Gamma(1/4)^(n + 1/4) * exp(3*n^2/4 + n/4 + 1/96 - C/(4*Pi))), where A = A074962 is the Glaisher-Kinkelin constant and C = A006752 = is Catalan's constant.