cp's OEIS Frontend

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

%I A079679 #59 Aug 14 2025 09:11:58
%S A079679 1,12,168,2424,35400,520236,7674144,113482584,1681028136,24932533800,
%T A079679 370144424376,5499182587416,81748907485248,1215834858032820,
%U A079679 18090048027643200,269246037610828656,4008495234662771688,59692297399976544120,889090275714779739120,13245013739104555683600
%N A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=6.
%C A079679 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.
%H A079679 Seiichi Manyama, <a href="/A079679/b079679.txt">Table of n, a(n) for n = 0..851</a>
%H A079679 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.
%H A079679 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.
%F A079679 a(n) = 3/5*(46656/3125)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.388...
%F A079679 c = 8/(3*sqrt(15*Pi)) = 0.388461664210517... - _Vaclav Kotesovec_, May 25 2020
%F A079679 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
%F A079679 From _Rui Duarte_ and António Guedes de Oliveira, Feb 17 2013: (Start)
%F A079679 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(6*n+1,k).
%F A079679 a(n) = Sum_{k=0..n} 6^(n-k) * binomial(5*n+k,k). (End)
%F A079679 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
%F A079679 a(n) = [x^n] 1/((1-6*x) * (1-x)^(5*n+1)). - _Seiichi Manyama_, Aug 03 2025
%F A079679 From _Seiichi Manyama_, Aug 14 2025: (Start)
%F A079679 a(n) = Sum_{k=0..n} 6^k * (-5)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
%F A079679 G.f.: g^2/(6-5*g)^2 where g = 1+x*g^6 is the g.f. of A002295. (End)
%o A079679 (PARI) a(n) = sum(k=0,n,5^(n-k)*binomial(6*n+1,k));
%o A079679 vector(30, n, a(n-1)) \\  _Altug Alkan_, Sep 30 2015
%Y A079679 Cf. A000302, A006256, A078995, A079563, A079678.
%Y A079679 Cf. A002295, A226705.
%K A079679 nonn
%O A079679 0,2
%A A079679 _Benoit Cloitre_, Jan 26 2003