OFFSET
0,2
COMMENTS
Equals the central terms of triangle A396840.
FORMULA
a(n) = Product_{k=1..n} (n+k)^(n+k) / k^k for n >= 0.
a(n) ~ 2^(2*n^2 + n + 1/12) * n^(n^2 - 1/12) / (A * exp(n^2/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jul 06 2026
EXAMPLE
The initial terms are
a(0) = 1;
a(1) = 2^2/1^1 = 4;
a(2) = (3^3*4^4)/(1^1*2^2)= 1728,
a(3) = (4^4*5^5*6^6)/(1^1*2^2*3^3) = 345600000;
a(4) = (5^5*6^6*7^7*8^8)/(1^1*2^2*3^3*4^4) = 72861813964800000;
...
MATHEMATICA
Table[Hyperfactorial[2*n]/Hyperfactorial[n]^2, {n, 0, 10}] (* Vaclav Kotesovec, Jul 06 2026 *)
PROG
(PARI) {H(n) = prod(k=1, n, k^k)} \\ hyperfactorial of n
{a(n) = H(2*n)/H(n)^2}
for(n=0, 10, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 06 2026
STATUS
approved