A079678 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=5.

1, 10, 115, 1360, 16265, 195660, 2361925, 28577440, 346316645, 4201744870, 51023399190, 620022989200, 7538489480075, 91696845873760, 1115794688036920, 13581508654978560, 165357977228808925, 2013721466517360650, 24527742112263770425, 298805688708113438240, 3640695209795092874290

Offset: 0

Author

Benoit Cloitre, Jan 26 2003

Keywords

• nonn

Comments

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.

Links

• Robert Israel, Table of n, a(n) for n = 0..828
• D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.
• Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.

Crossrefs

Cf. A006256, A078995, A079563, A079679.

Cf. A371753, A385632, A386812.

Programs

Maple

seq(add(binomial(5*k,k)*binomial(5*(n-k),n-k),k=0..n), n=0..30); # Robert Israel, Jul 16 2015

Mathematica

m = 5; Table[Sum[Binomial[m k, k] Binomial[m (n - k), n - k], {k, 0, n}], {n, 0, 17}] (* Michael De Vlieger, Sep 30 2015 *)

PARI

main(size)=my(k,n,m=5); concat(1,vector(size,n, sum(k=0,n, binomial(m*k,k)*binomial(m*(n-k),n-k)))) \\ Anders Hellström, Jul 16 2015

PARI

a(n) = sum(k=0,n,4^(n-k)*binomial(5*n+1,k));
vector(30, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

Formula

a(n) = 5/8*(3125/256)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.356...

c = sqrt(2)/sqrt(5*Pi) = 0.3568248232305542229... - Vaclav Kotesovec, May 25 2020

a(n) = Sum_{k=0..n} binomial(5*k+l,k) * binomial(5*(n-k)-l,n-k) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013

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

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(5*n+1,k).

a(n) = Sum_{k=0..n} 5^(n-k) * binomial(4*n+k,k). (End)

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/4, 1/2, 3/4], (3125/256)*x)^2 satisfies

((3125/2)*g^3*x^4-128*g^3*x^3)*g''''+((-3125*g^2*x^4+256*g^2*x^3)*g'+12500*g^3*x^3-576*g^3*x^2)*g'''+(-(9375/4)*g^2*x^4+192*g^2*x^3)*g''^2+(((28125/4)*g*x^4-576*g*x^3)*(g')^2+(-18750*g^2*x^3+864*g^2*x^2)*g'+22500*g^3*x^2-408*g^3*x)*g''+(-(46875/16)*x^4+240*x^3)*(g')^4+(9375*g*x^3-432*g*x^2)*(g')^3+(-11250*g^2*x^2+204*g^2*x)*(g')^2+(7500*g^3*x-12*g^3)*g'+120*g^4 = 0. - Robert Israel, Jul 16 2015

a(n) = [x^n] 1/((1-5*x) * (1-x)^(4*n+1)). - Seiichi Manyama, Aug 03 2025

From Seiichi Manyama, Aug 14 2025: (Start)

a(n) = Sum_{k=0..n} 5^k * (-4)^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).

G.f.: g^2/(5-4*g)^2 where g = 1+x*g^5 is the g.f. of A002294. (End)