A292797 2-Hankel transform of ((2*n - 1)!!, 2^n * n).

1, 1, 2, 24, 1728, 1658880, 17915904000, 4334215495680000, 18349334722510848000000, 2556904928296218824540160000000, 8768179290592246332614309314560000000000, 1343991010969776301093243630262125854720000000000000

Offset

0,3

Links

Table of n, a(n) for n=0..11.

Paul Barry, A Note on d-Hankel Transforms, Continued Fractions, and Riordan Arrays, arXiv:1702.04011 [math.CO], 2017.

Formula

a(n) = Prod_{k=0..n} ((k+2)*(k+1)^2)^(floor((n-k)/2)).

a(n) ~ 2^(3*n/4 + (5 + (-1)^n)/8) * Pi^(3*n/4 + (5 - (-1)^n)/8) * n^(3*n^2/4 + 5*n/4 + 3/8) / (A^(3/2) * exp(9*n^2/8 + 5*n/4 - 1/8)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jan 23 2024

Mathematica

f[n_]:=Product[((k + 2) (k + 1)^2)^(Floor[(n - k) / 2]), {k, 0, n}]; Table[f[n], {n, 0, 12}] (* Vincenzo Librandi, Sep 26 2017 *)

Prog

(Magma) [&*[((k+2)*(k+1)^2)^(Floor((n-k)/2)): k in [0..n]]: n in [0..16]]; // Vincenzo Librandi, Sep 26 2017

Crossrefs

Sequence in context: A181231 A111427 A081955 * A163086 A280922 A296589

Adjacent sequences: A292794 A292795 A292796 * A292798 A292799 A292800

Keyword

nonn

Author

Eric M. Schmidt, Sep 23 2017

Status

approved