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

1, 14, 231, 3934, 67851, 1177974, 20531770, 358788696, 6281076123, 110103674128, 1931983053056, 33926800240578, 596145343139514, 10480467311987778, 184327560283768776, 3243034966775972144, 57074433199551436347

Offset: 0

Author

Benoit Cloitre, Jan 26 2003

Keywords

• nonn

Comments

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

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..802
• Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.
• D. Merlini, R. Sprugnoli, and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.

Crossrefs

Cf. A006256, A078995, A079678, A079679.

Cf. A002296.

Formula

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

c = 10/(3*sqrt(21*Pi)) = 0.410387535383... - Vaclav Kotesovec, May 25 2020

From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)

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

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

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

a(n) = [x^n] 1/((1-7*x) * (1-x)^(6*n+1)). - Seiichi Manyama, Aug 03 2025

From Seiichi Manyama, Aug 14 2025: (Start)

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

G.f.: g^2/(7-6*g)^2 where g = 1+x*g^7 is the g.f. of A002296. (End)