OFFSET
0,5
LINKS
Robert Israel, Table of n, a(n) for n = 0..3087
FORMULA
G.f.: (1-x^3-sqrt((1-x^3)*(1-4*x^2-x^3)))/(2*x^2*(1-x^3)).
a(n) = Sum_{k=0..n/3}(((-1)^(n-3*k)+1)*(binomial((n-k)/2,k)*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2)))). - Vladimir Kruchinin, Apr 02 2016
(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6) = 0. - Robert Israel, Nov 04 2019
EXAMPLE
For n=6 we have 6 paths: UDUDUD, H3H3, UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).
MAPLE
f:= gfun:-rectoproc({(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 2}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Nov 04 2019
PROG
(Maxima)
a(n):=sum(((-1)^(n-3*k)+1)*((binomial((n-k)/2, k) )*(binomial(n-3*k, (n-3*k)/2))/((n-3*k+2))), k, 0, (n)/3); /* Vladimir Kruchinin, Apr 02 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
José Luis Ramírez Ramírez, Apr 21 2015
STATUS
approved