|
%I #7 Jul 26 2017 03:17:05
%S 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,
%T 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,
%U 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
%N 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).
%C Column 0 yields A109078.
%C T(2n,1)=0, T(2n-1,1) = A000957(n) (the Fine numbers).
%F 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).
%e T(5,2)=2 because we have uduududdud and uduuudddud, where u=(1,1), d=(1,-1).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 1, 0, 1;
%e 2, 0, 0, 1;
%e 4, 0, 1, 0, 1;
%e 6, 1, 2, 0, 0, 1;
%p 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
%Y Cf. A000957, A109078.
%K nonn,tabl
%O 0,7
%A _Emeric Deutsch_, Jun 17 2005
|
