login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A215340 Expansion of series_reversion( x/(1 + sum(k>=1, x^A032766(k)) ) ) / x. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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]

LINKS

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

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

EXAMPLE

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

MAPLE

b:= proc(x, y, t) option remember;

      `if`(y<x, 0, `if`(y=0, `if`(t=2, 0, 1),

      `if`(x>0 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[y<x, 0, If[y==0, If[t==2, 0, 1], If[x>0 && 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-Fran├žois Alcover, Apr 08 2015, after Alois P. Heinz *)

PROG

(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)

CROSSREFS

Cf. A215341.

Sequence in context: A265278 A111281 A018021 * A074405 A068786 A276359

Adjacent sequences:  A215337 A215338 A215339 * A215341 A215342 A215343

KEYWORD

nonn,easy

AUTHOR

Joerg Arndt, Aug 19 2012

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 22 01:24 EDT 2018. Contains 316431 sequences. (Running on oeis4.)