cp's OEIS Frontend

A114463 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an odd level (0<=k<=floor(n/2)-1 for n>=2; k=0 for n=0,1).

%I A114463 #13 Apr 06 2016 12:28:15
%S A114463 1,1,2,5,13,1,36,6,105,26,1,317,104,8,982,402,45,1,3105,1522,225,10,
%T A114463 9981,5693,1052,69,1,32520,21144,4698,412,12,107157,78188,20319,2249,
%U A114463 98,1,356481,288340,85864,11522,679,14,1195662,1061520,356535,56360,4230
%N A114463 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an odd level (0<=k<=floor(n/2)-1 for n>=2; k=0 for n=0,1).
%C A114463 Row n (n>=2) has floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k),k=0..floor(n/2)-1)=binomial(2n-4,n) (A002694). Column 0 yields A114465.
%H A114463 Alois P. Heinz, <a href="/A114463/b114463.txt">Rows n = 0..200, flattened</a>
%F A114463 G.f.: G=G(t, z) satisfies z[(1-t)z^2-(1-t)z+1]G^2-[1-(1-t)z^2]G+1=0.
%e A114463 T(5,1) = 6 because we have UUD(UU)DUDDD, UUD(UU)DDUDD, UUD(UU)DDDUD,
%e A114463 UDUUD(UU)DDD, UUDUD(UU)DDD and UUUDD(UU)DDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an odd level are shown between parentheses; note that the fourth path has an ascent of length 2 that starts at an even level).
%e A114463 Triangle starts:
%e A114463 :  0 :    1;
%e A114463 :  1 :    1;
%e A114463 :  2 :    2;
%e A114463 :  3 :    5;
%e A114463 :  4 :   13,    1;
%e A114463 :  5 :   36,    6;
%e A114463 :  6 :  105,   26,    1;
%e A114463 :  7 :  317,  104,    8;
%e A114463 :  8 :  982,  402,   45,  1;
%e A114463 :  9 : 3105, 1522,  225, 10;
%e A114463 : 10 : 9981, 5693, 1052, 69, 1;
%p A114463 G:=-1/2*(1-z^2+z^2*t-sqrt((z^2*t-z^2+4*z-1)*(z^2*t-z^2-1)))/z/(-z^2+z^2*t+z-z*t-1): Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 15 do P[n]:=coeff(Gser,z^n) od: 1; 1; for n from 2 to 15 do seq(coeff(t*P[n],t^j),j=1..floor(n/2)) od; # yields sequence in triangular form
%p A114463 # second Maple program:
%p A114463 b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
%p A114463      `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 2, 5, 2][t])
%p A114463       *`if`(t=5, z, 1) +b(x-1, y-1, [1, 3, 4, 1, 3][t]))))
%p A114463     end:
%p A114463 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
%p A114463 seq(T(n), n=0..15);  # _Alois P. Heinz_, Jun 10 2014
%t A114463 b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y+1, {2, 2, 2, 5, 2}[[t]]]*If[t==5, z, 1] + b[x-1, y-1, {1, 3, 4, 1, 3}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Mar 31 2015, after _Alois P. Heinz_ *)
%Y A114463 Cf. A114462, A114464, A114465, A000108, A002694, A243752.
%K A114463 nonn,tabf
%O A114463 0,3
%A A114463 _Emeric Deutsch_, Nov 29 2005