A109077 Triangle read by rows: T(n,k) is the number of symmetric Dyck paths of semilength n and having k hills (i.e., peaks at level 1).
1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 0, 1, 0, 1, 6, 1, 2, 0, 0, 1, 13, 0, 5, 0, 1, 0, 1, 22, 2, 6, 2, 2, 0, 0, 1, 46, 0, 16, 0, 6, 0, 1, 0, 1, 80, 6, 24, 4, 6, 3, 2, 0, 0, 1, 166, 0, 58, 0, 19, 0, 7, 0, 1, 0, 1, 296, 18, 90, 13, 26, 6, 6, 4, 2, 0, 0, 1, 610, 0, 211, 0, 71, 0, 22, 0, 8, 0, 1, 0, 1, 1106
Offset: 0
Examples
T(5,2)=2 because we have uduududdud and uduuudddud, where u=(1,1), d=(1,-1). Triangle begins: 1; 0, 1; 1, 0, 1; 2, 0, 0, 1; 4, 0, 1, 0, 1; 6, 1, 2, 0, 0, 1;
Programs
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Maple
G:=-2*(z+z*sqrt(1-4*z^2)-2*z^2-z*t-1-sqrt(1-4*z^2)+2*z^2*t-z*t*sqrt(1-4*z^2))/(-1-sqrt(1-4*z^2)+2*z)/(-1-sqrt(1-4*z^2)-2*z^2+2*z^2*t^2): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
Formula
G.f.: 2(1 + (t-1)z(1-2z) + q(1 - z + tz))/((1-2z+q)(1+2z^2-2t^2*z^2+q)), where q = sqrt(1 - 4z^2).
Comments