A187365 - OEIS

%I #19 Mar 20 2023 06:21:45

%S 1,42,2145,117572,6686100,388934370,22974421470,1372238454600,

%T 82653088824684,5011211083256840,305437356823765089,

%U 18697712969443807572,1148770108115543559100,70797430141465286938140,4374750896947475198160300,270950190057528375091435920

%N Trisection of A000984 (central binomial coefficients): binomial(2(3n+2),3n+2)/3!, n>=0.

%C See a comment under A187357 concerning trisection.

%C This appears also in the trisection of A001700: binomial(2*(3*n+1)+1,(3*n+1)+1)/3.

%H Seiichi Manyama, <a href="/A187365/b187365.txt">Table of n, a(n) for n = 0..554</a>

%F a(n)=binomial(2*(3*n+2),3*n+2)/3!, n>=0.

%F a(n)=binomial(3*(2*n+1),3*n+2)/3, n>=0.

%F O.g.f.:(cb(x^(1/3)) - sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))-(1-4*x^(1/3))/2))/(18*x^(2/3)),

%F with cb(x):=1/sqrt(1-4*x) (o.g.f. of A000984) and P(x):=P(-1/2,4*x)=1/sqrt(1+4*x+16*x^2) (o.g.f. of A116091, with P(x,z)the o.g.f. of the Legendre polynomials).

%F From _Peter Bala_, Mar 19 2023: (Start)

%F a(n) = (1/6)*Sum_{k = 0..3*n+2} binomial(3*n+2,k)^2.

%F a(n) = (1/6)*hypergeom([-2 - 3*n, -2 - 3*n], [1], 1).

%F a(n) = 8*(2*n + 1)*(6*n + 1)*(6*n - 1)/(n*(3*n + 1)*(3*n + 2)) * a(n-1). (End)

%Y Cf. A066802 binomial(6n,3n), A187364 binomial(2*(3n+1),3n+1)/2, A002458, A100033.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Mar 10 2011