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A114588 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at even levels (0<=k<=n-1; n>=1). A hill in a Dyck path is a peak at level 1.

%I A114588 #6 Jun 25 2013 15:48:46
%S A114588 0,0,1,1,0,1,1,3,1,1,3,6,6,2,1,7,17,18,11,3,1,17,48,58,40,18,4,1,43,
%T A114588 134,186,150,76,27,5,1,110,380,590,540,325,130,38,6,1,286,1083,1860,
%U A114588 1915,1305,624,206,51,7,1,753,3100,5844,6660,5115,2772,1097,308,66,8,1,2003
%N A114588 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k peaks at even levels (0<=k<=n-1; n>=1). A hill in a Dyck path is a peak at level 1.
%C A114588 Row n has n terms. Row sums are the Fine numbers (A000957). Column 0 yields A114589. Sum(k*T(n,k), k=0..n-1) yields A114590.
%F A114588 G.f.: G-1, where G = G(t,z) satisfies z(2+2z+z^2-tz-tz^2)G^2+(1+2z)(1+z-tz)G+1+z-tz=0.
%e A114588 T(4,3) = 1 because we have U(UD)(UD)(UD)D, where U=(1,1), D=(1,-1) (the peaks at even levels are shown between parentheses).
%e A114588 Triangle begins:
%e A114588 0;
%e A114588 0,   1;
%e A114588 1,   0,  1;
%e A114588 1,   3,  1,  1;
%e A114588 3,   6,  6,  2,  1;
%e A114588 7,  17, 18, 11,  3, 1;
%e A114588 17, 48, 58, 40, 18, 4, 1;
%p A114588 G:=(1-t*z+2*z^2+3*z-2*t*z^2-sqrt(1-3*z^2-2*z*t+2*z^2*t+z^2*t^2-2*z))/2/z/(2+2*z-t*z-t*z^2+z^2)-1: Gser:=simplify(series(G, z=0, 15)): for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 1 to 12 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form
%Y A114588 Cf. A000957, A114586, A114587, A114589, A114590.
%K A114588 nonn,tabl
%O A114588 1,8
%A A114588 _Emeric Deutsch_, Dec 11 2005