This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066802 #80 Feb 28 2025 06:24:25
%S A066802 1,20,924,48620,2704156,155117520,9075135300,538257874440,
%T A066802 32247603683100,1946939425648112,118264581564861424,
%U A066802 7219428434016265740,442512540276836779204,27217014869199032015600,1678910486211891090247320,103827421287553411369671120,6435067013866298908421603100
%N A066802 a(n) = binomial(6*n,3*n).
%C A066802 For the trisection of a sequence (here A000984) given by its real o.g.f. see a comment and a reference under A187357.
%H A066802 Harry J. Smith, <a href="/A066802/b066802.txt">Table of n, a(n) for n = 0..100</a>
%H A066802 R. J. Mathar, <a href="https://vixra.org/abs/2502.0097">The Eggenberger-Polya urn process: Probabilities of revisited ball ratios</a>, vixra:2502.0097 (2025) Table 1
%F A066802 a(n) = A000984(3*n).
%F A066802 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} binomial(n, i)*binomial(n, j) *binomial(n, k)*binomial(3n, i+j+k). - _Benoit Cloitre_, Mar 08 2005
%F A066802 O.g.f. (with a(0):=1): (cb(x^(1/3)) + sqrt(2)*P(x^(1/3))*sqrt(1/P(x^(1/3))+1+2*x^(1/3)))/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). - _Wolfdieter Lang_, Mar 24 2011
%F A066802 D-finite with recurrence n*(3n-1)*(3n-2)*a(n) = 8*(6n-5)*(6n-1)*(2n-1)*a(n-1). - _R. J. Mathar_, Sep 17 2012
%F A066802 a(n) = GegenbauerC(3*n, -3*n, -1). - _Peter Luschny_, May 07 2016
%F A066802 a(n) = hypergeom([-3*n, -3*n], [1], 1). - _Peter Luschny_, Mar 19 2018
%F A066802 a(n) ~ 2^(6*n)/sqrt(3*Pi*n). - _Vaclav Kotesovec_, Jun 07 2019
%F A066802 From _Peter Bala_, Feb 16 2020: (Start)
%F A066802 a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k.
%F A066802 a(n) = [(x*y)^(3*n)] (1 + x + y)^(6*n). Cf. A001448. (End)
%F A066802 Conjecture: a(n) = [x^n] G(x)^(2*n), where G(x) = (1 + x)*(1 - 6*x + x^2)/(2*x) + (x^2 - 1)*sqrt(1 - 14*x + x^2)/(2*x) = 1 + 10*x + 81*x^2 + 720*x^3 + .... The algebraic function G(x) satisfies the quadratic equation x*G(x)^2 - (1 - 5*x - 5*x^2 + x^3)*G(x) + (1 + x)^4 = 0. Cf. A001450. - _Peter Bala_, Oct 27 2022
%F A066802 a(n) = Sum_{k = 0..3*n} binomial(3*n+k-1, k). - _Peter Bala_, Jun 04 2024
%F A066802 O.g.f: 3F2(1/6,1/2,5/6; 1/3,2/3 ; 64*x). - _R. J. Mathar_, Jan 11 2025
%p A066802 a := n -> hypergeom([-3*n, -3*n], [1], 1):
%p A066802 seq(simplify(a(n)), n=0..13); # _Peter Luschny_, Mar 19 2018
%t A066802 Table[Binomial[6n, 3n], {n,0,13}] (* _Jean-François Alcover_, Jun 03 2019 *)
%o A066802 (PARI) a(n) = { binomial(6*n, 3*n) } \\ _Harry J. Smith_, Mar 28 2010
%o A066802 (Magma) [Binomial(6*n, 3*n): n in [0..15]]; // _G. C. Greubel_, Feb 17 2020
%o A066802 (Sage) [binomial(6*n, 3*n) for n in (0..15)] # _G. C. Greubel_, Feb 17 2020
%Y A066802 Cf. A001450, A066798, A187364 (binomial(2(3n+1),3n+1)/2), A187365 (binomial(2(3n+2),3n+2)/3!).
%K A066802 nonn,easy
%O A066802 0,2
%A A066802 _Benoit Cloitre_, Jan 18 2002
%E A066802 Extended to a(0)=1 by _M. F. Hasler_, Oct 06 2014