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A121531 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an even level (n >= 1, k >= 0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.

%I A121531 #8 Nov 26 2019 04:36:32
%S A121531 1,2,4,1,7,6,12,20,2,20,51,18,33,115,80,5,54,240,262,54,88,477,725,
%T A121531 294,13,143,916,1803,1158,161,232,1716,4170,3768,1026,34,376,3155,
%U A121531 9152,10815,4684,475,609,5717,19311,28418,17432,3449,89,986,10240,39520
%N A121531 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an even level (n >= 1, k >= 0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
%C A121531 Row n contains ceiling(n/2) terms.
%C A121531 Row sums are the odd-indexed Fibonacci numbers (A001519).
%C A121531 T(n,0) = Fibonacci(n+2) - 1 = A000071(n+2).
%C A121531 Sum_{k>=0} k*T(n,k) = A121532(n).
%H A121531 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 A121531 G.f.: G = G(t,z) = z(1 - 2tz^2 - tz^3)(1-tz^2)/((1 - z - tz^2)(1 - z - z^2 - 3tz^2 - tz^3 + t^2*z^4)).
%e A121531 T(5,2)=2 because we have UU/UU/UDDDDD and UU/UDDU/UDDD, where U=(1,1) and D=(1,-1) (the double rises at an even level are indicated by a /).
%e A121531 Triangle starts:
%e A121531    1;
%e A121531    2;
%e A121531    4,   1;
%e A121531    7,   6;
%e A121531   12,  20,  2;
%e A121531   20,  51, 18;
%e A121531   33, 115, 80, 5;
%p A121531 G:=z*(1-2*t*z^2-t*z^3)*(1-t*z^2)/(1-z-t*z^2)/(1-z-z^2-3*t*z^2-t*z^3+t^2*z^4): Gser:=simplify(series(G,z=0,18)): for n from 1 to 15 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 15 do seq(coeff(P[n],t,j),j=0..ceil(n/2)-1) od; # yields sequence in triangular form
%Y A121531 Cf. A001519, A121529, A121532.
%K A121531 nonn,tabf
%O A121531 1,2
%A A121531 _Emeric Deutsch_, Aug 05 2006