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A079679 a(n) = a(n,m) = sum(k=0,n,binomial(m*k,k)*binomial(m*(n-k),n-k)) for m=6. 0
1, 12, 168, 2424, 35400, 520236, 7674144, 113482584, 1681028136, 24932533800, 370144424376, 5499182587416, 81748907485248, 1215834858032820, 18090048027643200, 269246037610828656, 4008495234662771688 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..16.

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...

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

Sequence in context: A099745 A268899 A182606 * A216702 A320761 A282045

Adjacent sequences:  A079676 A079677 A079678 * A079680 A079681 A079682

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jan 26 2003

STATUS

approved

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Last modified July 23 19:33 EDT 2019. Contains 325263 sequences. (Running on oeis4.)