OFFSET
0,2
COMMENTS
a(n) is asymptotic to c*(46656/3125)^n/sqrt(n), with c = sqrt(3/(5*Pi)) = 0.437019372236831628217354... - Benoit Cloitre, Jan 23 2008
REFERENCES
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.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
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
G.f.: A(x) = x*B'(x)/B(x), where B(x)+1 is g.f. of A002295. - Vladimir Kruchinin, Oct 05 2015
a(n) = GegenbauerC(n, -3*n, -1). - Peter Luschny, May 07 2016
From Ilya Gutkovskiy, Jan 16 2017: (Start)
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).
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)
RHS of identities Sum_{k = 0..n} binomial(3*n, k)*binomial(3*n, n-k) =
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
From Peter Bala, Feb 20 2022: (Start)
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).
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 5*A(x))^5 + (6^6)*x*A(x)^6 = 0.
Sum_{n >= 1} a(n)*( x*(5*x + 6)^5/(6^6*(1 + x)^6) )^n = x. (End)
MATHEMATICA
Table[Binomial[6 n, n], {n, 0, 16}] (* Michael De Vlieger, Oct 05 2015 *)
PROG
(Magma) [Binomial(6*n, n): n in [0..100]]; // Vincenzo Librandi, Apr 13 2011
(Maxima)
B(x):=sum(binomial(6*n, n-1)/n*x^n, n, 1, 30);
taylor(x*diff(B(x), x)/B(x), x, 0, 10); /* Vladimir Kruchinin, Oct 05 2015 */
(PARI) a(n) = binomial(6*n, n) \\ Altug Alkan, Oct 05 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved