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%I #57 Sep 08 2022 08:44:32
%S 1,6,66,816,10626,142506,1947792,26978328,377348994,5317936260,
%T 75394027566,1074082795968,15363284301456,220495674290430,
%U 3173734438530120,45795673964460816,662252084388541314
%N Binomial coefficient C(6n,n).
%C a(n) is asymptotic to c*(46656/3125)^n/sqrt(n), with c = sqrt(3/(5*Pi)) = 0.437019372236831628217354... - _Benoit Cloitre_, Jan 23 2008
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%H T. D. Noe, <a href="/A004355/b004355.txt">Table of n, a(n) for n = 0..100</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F a(n) = C(6*n-1,n-1)*C(36*n^2,2)/(3*n*C(6*n+1,3)), n>0. - _Gary Detlefs_, Jan 02 2014
%F G.f.: A(x) = x*B'(x)/B(x), where B(x)+1 is g.f. of A002295. - _Vladimir Kruchinin_, Oct 05 2015
%F a(n) = GegenbauerC(n, -3*n, -1). - _Peter Luschny_, May 07 2016
%F From _Ilya Gutkovskiy_, Jan 16 2017: (Start)
%F O.g.f.: 5F4(1/6,1/3,1/2,2/3,5/6; 1/5,2/5,3/5,4/5; 46656*x/3125).
%F E.g.f.: 5F5(1/6,1/3,1/2,2/3,5/6; 1/5,2/5,3/5,4/5,1; 46656*x/3125). (End)
%F RHS of identities Sum_{k = 0..n} binomial(3*n, k)*binomial(3*n, n-k) =
%F Sum_{k = 0..2*n} (-1)^(n+k)*binomial(6*n, k)*binomial(6*n, 2*n-k) = binomial(6*n,n). - _Peter Bala_, Oct 07 2021
%F From _Peter Bala_, Feb 20 2022: (Start)
%F 5*n*(5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*a(n) = 6*(6*n-1)*(6*n-2)*(6*n-3)(6*n-4)*(6*n-5)*a(n-1).
%F The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 5*A(x))^5 + (6^6)*x*A(x)^6 = 0.
%F Sum_{n >= 1} a(n)*( x*(5*x + 6)^5/(6^6*(1 + x)^6) )^n = x. (End)
%t Table[Binomial[6 n, n], {n, 0, 16}] (* _Michael De Vlieger_, Oct 05 2015 *)
%o (Magma) [Binomial(6*n,n): n in [0..100]]; // _Vincenzo Librandi_, Apr 13 2011
%o (Maxima)
%o B(x):=sum(binomial(6*n,n-1)/n*x^n,n,1,30);
%o taylor(x*diff(B(x),x)/B(x),x,0,10); /* _Vladimir Kruchinin_, Oct 05 2015 */
%o (PARI) a(n) = binomial(6*n,n) \\ _Altug Alkan_, Oct 05 2015
%Y binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).
%Y Cf. A002295
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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