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Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at even level (n>=1,0<=k<=n-1). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
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%I #4 Mar 06 2018 11:32:41

%S 1,1,1,2,2,1,4,6,2,1,8,13,10,2,1,16,34,23,13,2,1,33,74,75,32,16,2,1,

%T 66,178,180,124,40,19,2,1,136,390,497,321,180,48,22,2,1,274,895,1192,

%U 1004,488,244,56,25,2,1,562,1958,3033,2598,1701,682,317,64,28,2,1,1138,4374

%N Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at even level (n>=1,0<=k<=n-1). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

%C Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,0)=A121485(n). Sum(k*T(n,k),k=0..n-1)=A121486(n).

%H E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170, 1997, 211-217.

%F G.f.: G(t,z) = z(1-z^2)(1-2z^2-tz^3)/(1-4z^2-z-tz+2tz^4+4z^4-z^6 +2z^3+tz^3+t^2*z^3).

%e T(4,2)=2 because we have UDUU|DU|DD and UU|DDUU|DD, where U=(1,1) and D=(1,-1) (the peaks at even level are shown by a |).

%e Triangle starts:

%e 1;

%e 1,1;

%e 2,2,1;

%e 4,6,2,1;

%e 8,13,10,2,1;

%e 16,34,23,13,2,1;

%p G:=z*(1-z^2)*(1-2*z^2-t*z^3)/(1-4*z^2-z-t*z+2*t*z^4+4*z^4-z^6+2*z^3+t*z^3+z^3*t^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

%Y Cf. A001519, A121481, A121485, A121486.

%K nonn,tabl

%O 1,4

%A _Emeric Deutsch_, Aug 02 2006