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A105640 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k UUDD's, where U=(1,1) and D=(1,-1) (0<=k<=floor(n/2), n>=2). A hill in a Dyck path is a peak at level 1.

%I A105640 #9 Apr 06 2019 09:29:51
%S A105640 0,1,1,1,2,3,1,5,10,3,14,29,13,1,39,89,52,6,111,279,195,36,1,322,881,
%T A105640 722,185,10,947,2806,2637,867,80,1,2818,8997,9528,3846,520,15,8470,
%U A105640 28997,34163,16382,2976,155,1,25677,93858,121749,67696,15631,1246,21
%N A105640 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k UUDD's, where U=(1,1) and D=(1,-1) (0<=k<=floor(n/2), n>=2). A hill in a Dyck path is a peak at level 1.
%C A105640 Row n has 1+floor(n/2) terms. Row sums are the Fine numbers (A000957). T(n,0)=A105641(n). Sum(k*T(n,k),k=0..floor(n/2))=A116914(n).
%H A105640 E. Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%F A105640 G.f.: G-1, where G =G(t,z) satisfies z(2+z+z^2-tz^2)G^2-(1+2z+z^2-tz^2)G+1=0.
%e A105640 T(5,2)=3 because we have U(UUDD)(UUDD)D, (UUDD)U(UUDD)D and U(UUDD)D(UUDD) (the UUDD's are shown between parentheses).
%e A105640 Triangle starts:
%e A105640   0,1;
%e A105640   1,1;
%e A105640   2,3,1;
%e A105640   5,10,3;
%e A105640   14,29,13,1;
%e A105640   ...
%p A105640 G:=(1+2*z+z^2-t*z^2-sqrt(1-4*z+2*z^2-2*t*z^2+z^4-2*z^4*t+t^2*z^4))/2/z/(2+z+z^2-t*z^2)-1: Gser:=simplify(series(G,z=0,17)): for n from 2 to 14 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form
%Y A105640 Cf. A000957, A105641, A116914.
%K A105640 nonn,tabf
%O A105640 2,5
%A A105640 _Emeric Deutsch_, May 08 2006