OFFSET
0,3
COMMENTS
See Novelli-Thibon (2014) for precise definition.
Sum over all Dyck paths of semilength n of products over all peaks p of (1+x_p-y_p), where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 27 2015
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 28.
FORMULA
G.f.: 1/(1-x) = Sum_{n>=0} a(n) * x^n*(1-x)^n / Product_{k=1..n} (1 + 2*k*x). - Paul D. Hanna, Jun 14 2014
MAPLE
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (1+x-y), 1) +
b(x-1, y+1, true)))
end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..25); # Alois P. Heinz, May 27 2015
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x==0, 1, b[x-1, y-1, False]*If[t, 1+x-y, 1] + b[x-1, y+1, True]]]; a[n_] := b[2*n, 0, False]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
PROG
(PARI) {a(n)=if(n<0, 0, polcoeff(1/(1-x+x*O(x^n)) - sum(k=1, n-1, a(k)*x^k*(1-x)^k/prod(j=0, k, 1+2*j*x+x*O(x^n))), n))}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 14 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 14 2014
EXTENSIONS
Offset changed to 0 by Paul D. Hanna, Jun 14 2014
STATUS
approved