A376872 a(n) = n! * 2^(-n) * binomial(3*n - 1, 2*n) * binomial(2*n, n). Central terms of the Bessel triangle A132062.

1, 1, 15, 420, 17325, 945945, 64324260, 5237832600, 496939367925, 53835098191875, 6557114959770375, 886998823648938000, 131941075017779527500, 21404902093269001807500, 3761147082102981746175000, 711609027933884146376310000, 144234254849349142918648333125

Offset: 0

Author

Peter Luschny, Oct 26 2024

Keywords

• nonn

Crossrefs

Cf. A132062, A245066.

Programs

Maple

a := n -> ifelse(n = 0, 1, (3*n)! / (3* 2^n * n!^2)): seq(a(n), n = 0..16);
# Alternative:
gf := 1 + x*hypergeom([4/3, 5/3], [2], (27*x)/2): ser := series(gf, x, 18):
seq(ifelse(n=0, 1, (n-1)!)*coeff(ser, x, n), n = 0..16);
# Or:
a := proc(n) option remember; if n < 2 then return 1 fi;
a(n - 1)*(27*n^3 - 54*n^2 + 33*n - 6)/(2*n^2 - 2*n) end:

Mathematica

Table[n!2^(-n)Binomial[3n-1,2n]Binomial[2n,n],{n,0,16}] (* James C. McMahon, Oct 27 2024 *)

Formula

a(n) = binomial(2*n-2*k, n-k)*binomial(2*n-k-1, k-1)*(n-k)!/2^(n-k) = A132062(2*n, n).

a(n) = (3*n)! / (3 * 2^n * n!^2) for n >= 1, that is A245066(n) / 3 for n >= 1.

a(n) = (n-1)! * [x^n] (1 + x*hypergeom([4/3, 5/3], [2], (27*x)/2)) for n >= 1.

a(n) = a(n - 1)*(27*n^3 - 54*n^2 + 33*n - 6)/(2*n^2 - 2*n).