A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=6.

1, 12, 168, 2424, 35400, 520236, 7674144, 113482584, 1681028136, 24932533800, 370144424376, 5499182587416, 81748907485248, 1215834858032820, 18090048027643200, 269246037610828656, 4008495234662771688, 59692297399976544120, 889090275714779739120, 13245013739104555683600

Offset: 0

Author

Benoit Cloitre, Jan 26 2003

Keywords

• nonn

Comments

More generally : a(n,m)=sum(k=0,n,binomial(m*k,k)*binomial(m*(n-k),n-k)) is asymptotic to 1/2*m/(m-1)*(m^m/(m-1)^(m-1))^n. See A000302, A006256, A078995 for cases m=2,3 and 4.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..851
• D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.
• Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.

Crossrefs

Cf. A000302, A006256, A078995, A079563, A079678.

Cf. A002295, A226705.

Programs

PARI

a(n) = sum(k=0,n,5^(n-k)*binomial(6*n+1,k));
vector(30, n, a(n-1)) \\  Altug Alkan, Sep 30 2015

Formula

a(n) = 3/5*(46656/3125)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.388...

c = 8/(3*sqrt(15*Pi)) = 0.388461664210517... - Vaclav Kotesovec, May 25 2020

a(n) = Sum_{k=0..n} binomial(6*k+l,k)*binomial(6*(n-k)-l,n-k) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013

From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)

a(n) = Sum_{k=0..n} 5^(n-k) * binomial(6*n+1,k).

a(n) = Sum_{k=0..n} 6^(n-k) * binomial(5*n+k,k). (End)

G.f.: hypergeom([1/6, 1/3, 1/2, 2/3, 5/6],[1/5, 2/5, 3/5, 4/5],46656*x/3125)^2. - Mark van Hoeij, Apr 19 2013

a(n) = [x^n] 1/((1-6*x) * (1-x)^(5*n+1)). - Seiichi Manyama, Aug 03 2025

From Seiichi Manyama, Aug 14 2025: (Start)

a(n) = Sum_{k=0..n} 6^k * (-5)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).

G.f.: g^2/(6-5*g)^2 where g = 1+x*g^6 is the g.f. of A002295. (End)