A215340 Expansion of series_reversion( x/(1 + sum(k>=1, x^A032766(k)) ) ) / x.

1, 1, 1, 2, 6, 16, 40, 107, 307, 893, 2597, 7646, 22878, 69162, 210402, 644098, 1984598, 6149428, 19143220, 59840692, 187781992, 591343894, 1868106990, 5918537492, 18800935948, 59869902152, 191081899648, 611138052146, 1958410654202, 6287175115130, 20218209139666, 65120537016867

Offset: 0

Author

Joerg Arndt, Aug 19 2012

Keywords

• nonn
• easy

Comments

Number of Dyck n-paths avoiding ascents of length == 2 mod 3, see example. - David Scambler, Apr 16 2013

This is a special case of the following: let S be a set of positive numbers, r(x) = x/(1 + sum(e in S, x^e)), and f(x)=series_reversion(r(x)) / x, then f is the g.f. for the number of Dyck words of semilength n with substrings UUU...UU only of lengths e in S (that is, all ascent lengths are in S). [Joerg Arndt, Apr 16 2013]

Examples

			The 16 Dyck words of semilength 5 without substrings UUU..UU of length 2, 5, 8, etc. (using '1' for U and '.' for D) are
01:   1.1.1.1.1.
02:   1.1.111...
03:   1.111...1.
04:   1.111..1..
05:   1.111.1...
06:   1.1111....
07:   111...1.1.
08:   111..1..1.
09:   111..1.1..
10:   111.1...1.
11:   111.1..1..
12:   111.1.1...
13:   1111....1.
14:   1111...1..
15:   1111..1...
16:   1111.1....
- _Joerg Arndt_, Apr 16 2013
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..750

Crossrefs

Cf. A215341.

Programs

Maple

b:= proc(x, y, t) option remember;
      `if`(y0 and t<>2, b(x-1, y, 0), 0)+b(x, y-1, irem(t+1, 3))))
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 16 2013

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y0 && t != 2, b[x-1, y, 0], 0] + b[x, y-1, Mod[t+1, 3]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 08 2015, after Alois P. Heinz *)

PARI

N = 66;  x = 'x + O('x^N);
rf = x/(1+sum(n=1, N, ((n%3)!=2)*x^n ) );
gf = serreverse(rf)/x;
v = Vec(gf)

Formula

G.f. A(x) satisfies 0 = -x^3*A(x)^4 + (-x + 1)*A(x) - 1. [Joerg Arndt, Mar 01 2014]

Recurrence: 27*(n-1)*n*(n+1)*(2*n-5)*(4*n-11)*(4*n-7)*a(n) = 9*(n-1)*n*(4*n-11)*(96*n^3 - 456*n^2 + 616*n - 197)*a(n-1) - 3*(n-1)*(1728*n^5 - 15552*n^4 + 53164*n^3 - 85322*n^2 + 63369*n - 17010)*a(n-2) + (4*n-9)*(4*n-3)*(728*n^4 - 6188*n^3 + 19267*n^2 - 25987*n + 12810)*a(n-3) - 3*(n-3)*(2*n-3)*(3*n-10)*(3*n-8)*(4*n-7)*(4*n-3)*a(n-4). - Vaclav Kotesovec, Mar 22 2014

a(n) ~ sqrt(2*(3+r)/(3*(1-r)^3)) / (3*sqrt(Pi)*n^(3/2)*r^n), where r = 0.295932936709444136... is the root of the equation 27*(1-r)^4 = 256*r^3. - Vaclav Kotesovec, Mar 22 2014

a(n) = 1/(n + 1)*Sum_{k = 0..floor(n/3)} binomial(n + 1, n - 3*k)*binomial(n + k, n). - Peter Bala, Aug 02 2016

Extensions

Modified definition to obtain offset 0 for combinatorial interpretation, Joerg Arndt, Apr 16 2013