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

0,3

Links

Table of n, a(n) for n=0..16.

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).

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 *)

Crossrefs

Cf. A132062, A245066.

Sequence in context: A069431 A133791 A361284 * A323781 A253447 A302112

Adjacent sequences: A376869 A376870 A376871 * A376873 A376874 A376875

Keyword

nonn

Author

Peter Luschny, Oct 26 2024

Status

approved