A004355 Binomial coefficient C(6n,n).

1, 6, 66, 816, 10626, 142506, 1947792, 26978328, 377348994, 5317936260, 75394027566, 1074082795968, 15363284301456, 220495674290430, 3173734438530120, 45795673964460816, 662252084388541314

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

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).

Cf. A002295

Sequence in context: A127858 A173535 A267141 * A282046 A124862 A130977

Adjacent sequences: A004352 A004353 A004354 * A004356 A004357 A004358

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved