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

%I A376872 #24 Mar 30 2025 18:41:35
%S A376872 1,1,15,420,17325,945945,64324260,5237832600,496939367925,
%T A376872 53835098191875,6557114959770375,886998823648938000,
%U A376872 131941075017779527500,21404902093269001807500,3761147082102981746175000,711609027933884146376310000,144234254849349142918648333125
%N A376872 a(n) = n! * 2^(-n) * binomial(3*n - 1, 2*n) * binomial(2*n, n). Central terms of the Bessel triangle A132062.
%F A376872 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).
%F A376872 a(n) = (3*n)! / (3 * 2^n * n!^2) for n >= 1, that is A245066(n) / 3 for n >= 1.
%F A376872 a(n) = (n-1)! * [x^n] (1 + x*hypergeom([4/3, 5/3], [2], (27*x)/2)) for n >= 1.
%F A376872 a(n) = a(n - 1)*(27*n^3 - 54*n^2 + 33*n - 6)/(2*n^2 - 2*n).
%p A376872 a := n -> ifelse(n = 0, 1, (3*n)! / (3* 2^n * n!^2)): seq(a(n), n = 0..16);
%p A376872 # Alternative:
%p A376872 gf := 1 + x*hypergeom([4/3, 5/3], [2], (27*x)/2): ser := series(gf, x, 18):
%p A376872 seq(ifelse(n=0, 1, (n-1)!)*coeff(ser, x, n), n = 0..16);
%p A376872 # Or:
%p A376872 a := proc(n) option remember; if n < 2 then return 1 fi;
%p A376872 a(n - 1)*(27*n^3 - 54*n^2 + 33*n - 6)/(2*n^2 - 2*n) end:
%t A376872 Table[n!2^(-n)Binomial[3n-1,2n]Binomial[2n,n],{n,0,16}] (* _James C. McMahon_, Oct 27 2024 *)
%Y A376872 Cf. A132062, A245066.
%K A376872 nonn
%O A376872 0,3
%A A376872 _Peter Luschny_, Oct 26 2024