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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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].

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

Programs

Magma

[Binomial(6*n,n): n in [0..100]]; // Vincenzo Librandi, Apr 13 2011

Mathematica

Table[Binomial[6 n, n], {n, 0, 16}] (* Michael De Vlieger, Oct 05 2015 *)

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

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)