OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..39
R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.
FORMULA
From G. C. Greubel, Aug 29 2023: (Start)
a(n) = (2^(n+3)/Pi)^(n/2)*BarnesG(n+3/2)/(Gamma(n+ 2)*BarnesG(3/2)).
a(n) = (1/2)^binomial(n,2)*BarnesG(n+1)*Product_{k=2..n} binomial(2*k, k+1).
a(n) = Product_{k=1..n-1} (2*k+2)!/(2^k*(k+2)!). (End)
a(n) ~ sqrt(A/Pi) * 2^(n^2/2 + 2*n - 7/24) * n^(n^2/2 - n/2 - 35/24) / exp(3*n^2/4 - n/2 + 1/24), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023
MATHEMATICA
(* First program *)
f[j_]:= j*(j+1)/2; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j, k-1}], {k, 2, n}]
d[n_]:= Product[(i-1)!, {i, n}]
Table[v[n], {n, 0, z}] (* A203309 *)
Table[v[n+1]/v[n], {n, z}] (* A203310 *)
Table[v[n]/d[n], {n, 0, 12}] (* A203467 *)
(* Second program *)
Table[Product[(2*k+2)!/(2^k*(k+2)!), {k, n-1}], {n, 0, 20}] (* G. C. Greubel, Aug 29 2023 *)
PROG
(Magma) F:= Factorial; [1] cat [(&*[(F(2*k+2))/(2^k*F(k+2)): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 29 2023
(SageMath) f=factorial; [product((f(2*j+2))/(2^j*f(j+2)) for j in range(n)) for n in range(21)] # G. C. Greubel, Aug 29 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 02 2012
EXTENSIONS
Name edited by Michel Marcus, May 17 2019
a(0) = 1 prepended by G. C. Greubel, Aug 29 2023
STATUS
approved