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A143952 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.

%I A143952 #7 Jun 11 2017 08:34:38
%S A143952 1,1,1,1,1,4,1,12,1,1,33,8,1,88,42,1,1,232,183,13,1,609,717,102,1,1,
%T A143952 1596,2622,624,19,1,4180,9134,3275,205,1,1,10945,30691,15473,1650,26,
%U A143952 1,28656,100284,67684,11020,366,1,1,75024,320466,279106,64553,3716,34,1
%N A143952 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peak plateaux (0<=k<=floor(n/2)). A peak plateau is a run of consecutive peaks that is preceded by an upstep and followed by a down step; a peak consists of an upstep followed by a downstep.
%C A143952 Row n has 1+floor(n/2) terms.
%C A143952 Row sums are the Catalan numbers (A000108).
%C A143952 T(n,1)=A027941(n-1)=Fibonacci(2n-1)-1.
%C A143952 Sum(k*T(n,k),k=0..floor(n/2))=A079309(n-1).
%C A143952 For the statistic "number of peaks in peak plateaux", see A143953.
%H A143952 David Callan, <a href="http://www.stat.wisc.edu/~callan/notes/">Kreweras's Narayana Number Identity Has a Simple Dyck Path Interpretation .</a>
%F A143952 The g.f. G=G(t,z) satisfies z(1-z)G^2 - (1-z+z^2-tz^2)G+1-z = 0 (for the explicit form of G see the Maple program).
%F A143952 The trivariate g.f. g=g(x,y,z) of Dyck paths with respect to number of peak plateaux, number of peaks in the peak plateaux and semilength, marked, by x, y and z, respectively satisfies g=1+zg[g+xyz/(1-yz)-z/(1-z)].
%F A143952 T(n,k) = Sum_{r=1..n} Narayana(n-r,k)*binomial(2n-r-k,r-k) where Narayana(n,k) := binomial(n,k)*binomial(n,k-1)/n is the Narayana number A001263. - _David Callan_, Oct 31 2008
%e A143952 T(3,1)=4 because we have UD(UUDD), (UUDD)UD, (UUDUDD) and U(UUDD)D (the peak plateaux are shown between parentheses).
%e A143952 The triangle starts:
%e A143952 1;
%e A143952 1;
%e A143952 1,1;
%e A143952 1,4;
%e A143952 1,12,1;
%e A143952 1,33,8;
%e A143952 1,88,42,1;
%p A143952 C:=proc(z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G:=(1-z)*C(z*(1-z)^2/(1-z+z^2-t*z^2)^2)/(1-z+z^2-t*z^2): Gser:=simplify(series(G,z= 0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A143952 Cf. A000045, A000108, A027941, A079309, A143953.
%K A143952 nonn,tabf
%O A143952 0,6
%A A143952 _Emeric Deutsch_, Oct 10 2008