A114489 - OEIS

%I #15 May 20 2015 05:45:36

%S 1,1,2,4,1,9,4,1,22,14,5,1,58,46,21,6,1,163,149,80,29,7,1,483,484,292,

%T 124,38,8,1,1494,1589,1044,498,179,48,9,1,4783,5288,3701,1928,780,246,

%U 59,10,1,15740,17848,13096,7304,3237,1152,326,71,11,1,52956,61060,46428

%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n that have k valleys at level 1.

%C T(n,k) is also the number of Dyck paths of semilength n having k pairs of consecutive valleys at the same level. Example: T(4,1)=4 because we have U(DU)(DU)UDD, U(DU)UD(DU)D, UUD(DU)(DU)D, and UU(DU)(DU)DD, where U=(1,1), D=(1,-1); the pairs of consecutive same-level valleys are shown between parentheses. - _Emeric Deutsch_, Jun 19 2011

%C Rows 0 and 1 contain one term each; row n contains n-1 terms (n>=2).

%C Row sums are the Catalan numbers (A000108).

%C Column 0 yields A059019.

%C Sum(k*T(n,k), k=0..n-1) = 6*binomial(2*n-1,n-3)/(n+3) (A003517).

%H Alois P. Heinz, <a href="/A114489/b114489.txt">Rows n = 0..150, flattened</a>

%F G.f.: (1-t*z*C)/((1-z)*(1-t*z*C)-z^2*C), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.

%e T(4,1) = 4 because we have UU(DU)DDUD, UDUU(DU)DD, UU(DU)UDDD and UUUD(DU)DD, where U=(1,1), D=(1,-1); the valleys at level 1 are shown between parentheses.

%e Triangle starts:

%e 1;

%e 1;

%e 2;

%e 4,   1;

%e 9,   4, 1;

%e 22, 14, 5, 1;

%p C:=(1-sqrt(1-4*z))/2/z: G:=(1-t*z*C)/(1-t*z*C-z+t*z^2*C-z^2*C): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: 1; 1; for n from 2 to 12 do seq(coeff(t*P[n],t^j),j=1..n-1) od; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,

%p       `if`(x=0, 1, expand(b(x-1, y-1, 1)+

%p       `if`(t=1 and y=1, z, 1)*b(x-1, y+1, 0))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):

%p seq(T(n), n=0..14);  # _Alois P. Heinz_, Mar 12 2014

%t b[x_, y_, t_] :=  b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, Expand[b[x-1, y-1, 1] + If[t == 1 && y == 1, z, 1]*b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, May 20 2015, after _Alois P. Heinz_ *)

%Y Cf. A000108, A059019, A003517.

%K nonn,tabf

%O 0,3

%A _Emeric Deutsch_, Dec 01 2005