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A187364 Trisection of A000984 (central binomial coefficients): binomial(2(3n+1),3n+1)/2, n>=0.

%I A187364 #23 Nov 06 2024 04:41:46
%S A187364 1,35,1716,92378,5200300,300540195,17672631900,1052049481860,
%T A187364 63205303218876,3824345300380220,232714176627630544,
%U A187364 14226520737620288370,873065282167813104916,53753604366668088230810,3318776542511877736535400,205397724721029574666088520
%N A187364 Trisection of A000984 (central binomial coefficients): binomial(2(3n+1),3n+1)/2, n>=0.
%C A187364 See a comment under A187363 concerning trisection.
%C A187364 This appears also in the trisection of A001700 (central binomials in the odd numbered Pascal rows):  binomial(2*(3*n)+1,3*n+1).
%H A187364 Seiichi Manyama, <a href="/A187364/b187364.txt">Table of n, a(n) for n = 0..554</a>
%F A187364 a(n) = binomial(2*(3*n+1),3*n+1)/2, n>=0.
%F A187364 a(n) = binomial(2*(3*n)+1,3*n+1), n>=0.
%F A187364 O.g.f.: (cb(x^(1/3)) - sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))-(1+8*x^(1/3))/2))/(6*x^(1/3)), 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 A187364 From _Peter Bala_, Mar 19 2023: (Start)
%F A187364 a(n) = (1/2)*Sum_{k = 0..3*n+1} binomial(3*n+1,k)^2.
%F A187364 a(n) = (1/2)*hypergeom([-1 - 3*n, -1 - 3*n], [1], 1).
%F A187364 a(n) = 8*(2*n - 1)*(6*n + 1)*(6*n - 1)/(n*(3*n + 1)*(3*n - 1)) * a(n-1). (End)
%F A187364 Right-hand side of the binomial sum identity (1/18) * Sum_{k = 0..6*n+3} (-1)^(n+k) * (k/(2*n + 1))^2 * binomial(6*n+3, k)^2 = a(n). - _Peter Bala_, Nov 05 2024
%t A187364 Table[c=3n+1;Binomial[2c,c]/2,{n,0,20}] (* _Harvey P. Dale_, May 10 2012 *)
%Y A187364 Cf. A066802 (binomial(6n,3n)), A187365 (binomial(2(3n+2),3n+2)/3!).
%Y A187364 Cf. A002458, A100033.
%K A187364 nonn,easy
%O A187364 0,2
%A A187364 _Wolfdieter Lang_, Mar 10 2011