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A079678 a(n) = a(n,m) = sum(k=0,n,binomial(m*k,k)*binomial(m*(n-k),n-k)) for m=5. 1
1, 10, 115, 1360, 16265, 195660, 2361925, 28577440, 346316645, 4201744870, 51023399190, 620022989200, 7538489480075, 91696845873760, 1115794688036920, 13581508654978560, 165357977228808925, 2013721466517360650 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

FORMULA

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

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

a(n) = sum(k=0,n,4^(n-k)*binomial(5n+1,k)). - Rui Duarte and António Guedes de Oliveira, Feb 17 2013

a(n) = sum(k=0,n,5^(n-k)*binomial(4n+k,k)). - Rui Duarte and António Guedes de Oliveira, Feb 17 2013

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

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 *)

PROG

(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

CROSSREFS

Sequence in context: A185391 A104520 A138845 * A233908 A089833 A308667

Adjacent sequences:  A079675 A079676 A079677 * A079679 A079680 A079681

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jan 26 2003

STATUS

approved

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Last modified October 21 14:58 EDT 2019. Contains 328301 sequences. (Running on oeis4.)