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

Offset

0,2

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.

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

a(n) = sum(k=0,n,5^(n-k)*binomial(6n+1,k)). - Rui Duarte and António Guedes de Oliveira, Feb 17 2013

a(n) = sum(k=0,n,6^(n-k)*binomial(5n+k,k)) - Rui Duarte and António Guedes de Oliveira, Feb 17 2013

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

Prog

(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

Crossrefs

Cf. A000302, A006256, A078995.

Sequence in context: A268899 A366710 A182606 * A216702 A320761 A282045

Adjacent sequences: A079676 A079677 A079678 * A079680 A079681 A079682

Keyword

nonn

Author

Benoit Cloitre, Jan 26 2003

Status

approved