A079563 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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.

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)