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A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=7. 2

%I #39 Oct 31 2022 19:26:29

%S 1,14,231,3934,67851,1177974,20531770,358788696,6281076123,

%T 110103674128,1931983053056,33926800240578,596145343139514,

%U 10480467311987778,184327560283768776,3243034966775972144,57074433199551436347

%N a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=7.

%C More generally, for m>=2, 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 * (1 + (2*m-4)/(3*sqrt(Pi*n*m*(m-1)/2))), extended by _Vaclav Kotesovec_, May 25 2020

%C See A000302, A006256, A078995 for cases m=2,3 and 4.

%H Seiichi Manyama, <a href="/A079563/b079563.txt">Table of n, a(n) for n = 0..802</a>

%H Rui Duarte and António Guedes de Oliveira, <a href="http://arxiv.org/abs/1302.2100">Short note on the convolution of binomial coefficients</a>, arXiv:1302.2100 [math.CO], 2013.

%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344.

%F a(n) = (7/12)*(823543/46656)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.41...

%F c = 10/(3*sqrt(21*Pi)) = 0.410387535383... - _Vaclav Kotesovec_, May 25 2020

%F From _Rui Duarte_ and António Guedes de Oliveira, Feb 17 2013: (Start)

%F a(n) = Sum_{k=0..n} binomial(7*k+x,k)*binomial(7*(n-k)-x,n-k) for any real x.

%F a(n) = Sum_(k=0..n} 6^(n-k)*binomial(7n+1,k).

%F a(n) = Sum_{k=0..n} 7^(n-k)*binomial(6n+k,k). (End)

%Y Cf. A006256, A078995, A079678, A079679.

%K nonn

%O 0,2

%A _Benoit Cloitre_, Jan 26 2003

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)