OFFSET
0,2
COMMENTS
Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 0 and two kinds of (1,1). - Alois P. Heinz, Oct 07 2015
Number of pairs of noncrossing paths of length n which start and end together, each taking steps (1,0), (1,1) or (1,-1) (i.e., Motzkin-type). - Nicholas R. Beaton, Jun 17 2024
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = (1/x)*Series_Reversion[x*(1-x)^2/((1+x)^2*(1-x+x^2))].
G.f.: A(x) satisfies A(x^2) = M(x)*M(-x), where M(x) is the g.f. of A001006. - Alexander Burstein, Oct 03 2017
G.f.: A(x) satisfies A(x^2) = (1-x - sqrt(1-2*x-3*x^2))*(1+x - sqrt(1+2*x-3*x^2))/(4*x^4). - Paul D. Hanna, Oct 05 2017, concluded from formula of Alexander Burstein.
EXAMPLE
G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 509*x^4 + 3459*x^5 + ...
log(A(x)) = 3*x + 19*x^2/2 + 141*x^3/3 + 1107*x^4/4 + 8953*x^5/5 + ... + A082758(n)*x^n/n + ...
MAPLE
b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, Oct 07 2015
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 3, 14, 80][n+1],
((10*(n+1))*(16*n^3-20*n^2-n-1) *a(n-1)
+(-944*n^4+2596*n^3-1924*n^2+236*n+30) *a(n-2)
+(90*(n-2))*(16*n^3-52*n^2+45*n-6) *a(n-3)
-(81*(2*n-5))*(n-2)*(n-3)*(4*n-1) *a(n-4))/
((n+1)*(4*n-5)*(2*n+1)*(n+2)))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 07 2015
MATHEMATICA
(1/x)*InverseSeries[x*(1 - x)^2/((1 + x)^2*(1 - x + x^2)) + O[x]^30, x] // CoefficientList[#, x]& (* Jean-François Alcover, Jun 09 2018 *)
PROG
(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, polcoeff((1+x+x^2)^m, k)^2)*x^m/m) +x*O(x^n)), n))}
(PARI) {a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 01 2009
STATUS
approved