A215340 - OEIS

%I #32 Aug 03 2016 04:42:29

%S 1,1,1,2,6,16,40,107,307,893,2597,7646,22878,69162,210402,644098,

%T 1984598,6149428,19143220,59840692,187781992,591343894,1868106990,

%U 5918537492,18800935948,59869902152,191081899648,611138052146,1958410654202,6287175115130,20218209139666,65120537016867

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

%C Number of Dyck n-paths avoiding ascents of length == 2 mod 3, see example. - _David Scambler_, Apr 16 2013

%C 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]

%H Alois P. Heinz, <a href="/A215340/b215340.txt">Table of n, a(n) for n = 0..750</a>

%F G.f. A(x) satisfies 0 = -x^3*A(x)^4 + (-x + 1)*A(x) - 1. [_Joerg Arndt_, Mar 01 2014]

%F 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

%F 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

%F 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

%e 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

%e 01:   1.1.1.1.1.

%e 02:   1.1.111...

%e 03:   1.111...1.

%e 04:   1.111..1..

%e 05:   1.111.1...

%e 06:   1.1111....

%e 07:   111...1.1.

%e 08:   111..1..1.

%e 09:   111..1.1..

%e 10:   111.1...1.

%e 11:   111.1..1..

%e 12:   111.1.1...

%e 13:   1111....1.

%e 14:   1111...1..

%e 15:   1111..1...

%e 16:   1111.1....

%e - _Joerg Arndt_, Apr 16 2013

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

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

%p       `if`(x>0 and t<>2, b(x-1, y, 0), 0)+b(x, y-1, irem(t+1, 3))))

%p     end:

%p a:= n-> b(n, n, 0):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 16 2013

%t 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-François Alcover_, Apr 08 2015, after _Alois P. Heinz_ *)

%o (PARI)

%o N = 66;  x = 'x + O('x^N);

%o rf = x/(1+sum(n=1, N, ((n%3)!=2)*x^n ) );

%o gf = serreverse(rf)/x;

%o v = Vec(gf)

%Y Cf. A215341.

%K nonn,easy

%O 0,4

%A _Joerg Arndt_, Aug 19 2012

%E Modified definition to obtain offset 0 for combinatorial interpretation, _Joerg Arndt_, Apr 16 2013