This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079563 #53 Aug 14 2025 09:12:05
%S A079563 1,14,231,3934,67851,1177974,20531770,358788696,6281076123,
%T A079563 110103674128,1931983053056,33926800240578,596145343139514,
%U A079563 10480467311987778,184327560283768776,3243034966775972144,57074433199551436347
%N A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=7.
%C A079563 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 A079563 See A000302, A006256, A078995 for cases m=2,3 and 4.
%H A079563 Seiichi Manyama, <a href="/A079563/b079563.txt">Table of n, a(n) for n = 0..802</a>
%H A079563 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 A079563 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 A079563 a(n) = (7/12)*(823543/46656)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.41...
%F A079563 c = 10/(3*sqrt(21*Pi)) = 0.410387535383... - _Vaclav Kotesovec_, May 25 2020
%F A079563 From _Rui Duarte_ and António Guedes de Oliveira, Feb 17 2013: (Start)
%F A079563 a(n) = Sum_{k=0..n} binomial(7*k+x,k)*binomial(7*(n-k)-x,n-k) for any real x.
%F A079563 a(n) = Sum_{k=0..n} 6^(n-k)*binomial(7*n+1,k).
%F A079563 a(n) = Sum_{k=0..n} 7^(n-k)*binomial(6*n+k,k). (End)
%F A079563 a(n) = [x^n] 1/((1-7*x) * (1-x)^(6*n+1)). - _Seiichi Manyama_, Aug 03 2025
%F A079563 From _Seiichi Manyama_, Aug 14 2025: (Start)
%F A079563 a(n) = Sum_{k=0..n} 7^k * (-6)^(n-k) * binomial(7*n+1,k) * binomial(7*n-k,n-k).
%F A079563 G.f.: g^2/(7-6*g)^2 where g = 1+x*g^7 is the g.f. of A002296. (End)
%Y A079563 Cf. A006256, A078995, A079678, A079679.
%Y A079563 Cf. A002296.
%K A079563 nonn
%O A079563 0,2
%A A079563 _Benoit Cloitre_, Jan 26 2003