|
| |
|
|
A079678
|
|
a(n) = a(n,m) = sum(k=0,n,binomial(m*k,k)*binomial(m*(n-k),n-k)) for m=5.
|
|
3
|
|
|
|
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
|
|
|
|
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
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));
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
