OFFSET
0,3
COMMENTS
a(n) = A191518(n,0).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 2/(1-2*x-x^2+sqrt(1-2*x^2-3*x^4)).
a(n-1)=sum(m=floor((n+1)/2)..n, ((2*m-n)*sum(j=0..m, binomial(j,n+2*j-3*m)*binomial(m,j)))/m). [Vladimir Kruchinin, Mar 09 2013]
D-finite with recurrence (n+1)*a(n) +(-n-1)*a(n-1) +3*(-n+1)*a(n-2) +(n-5)*a(n-3) +(-n+11)*a(n-4) +(5*n-19)*a(n-5) +3*(n-5)*a(n-6) +3*(n-5)*a(n-7)=0. - R. J. Mathar, Jul 24 2022
EXAMPLE
a(6) = 19 because among the 20 (=A001405(6)) dispersed Dyck paths of length 6 only UUUDDD has a UUU (U=(1,1), D=(1,-1)).
MAPLE
g := 2/(1-2*z-z^2+sqrt(1-2*z^2-3*z^4)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 36);
# second Maple program:
a:= proc(n) option remember;
`if`(n<7, [1, 1, 2, 3, 6, 10, 19][n+1],
((n+1)*a(n-1)+(3*n-3)*a(n-2)-(n-5)*a(n-3)
+(n-11)*a(n-4)-(5*n-19)*a(n-5)-(3*n-15)*a(n-6)
-(3*n-15)*a(n-7))/(n+1))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Jun 02 2014
MATHEMATICA
a[n_] := Sum[((2*m-n-1)*Sum[Binomial[j, n+2*j-3*m+1]*Binomial[m, j], {j, 0, m}])/m, {m, Floor[(n+2)/2], n+1}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 31 2015, after Vladimir Kruchinin *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 07 2011
STATUS
approved