Search: id:A121529
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%I A121529
%S A121529 1,1,1,1,4,1,10,2,1,19,14,1,33,50,5,1,55,132,45,1,90,301,205,13,1,146,
%T A121529 631,680,139,1,236,1255,1892,763,34,1,381,2409,4717,3019,419,1,615,4509,
%U A121529 10920,9846,2677,89,1,993,8283,23974,28292,12241,1241,1,1604,14998
%N A121529 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths 
               of semilength n and having k double rises at an odd 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 A121529 Row n contains 1+floor(n/2) terms. Row sums are the odd-subscripted Fibonacci 
               numbers (A001519). T(2n,n)=Fibonacci(2n-1) (A001519). Sum(k*T(n,k), 
               k>=0)=A121530(n).
%D A121529 E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck 
               paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
%F A121529 G.f.=G(t,z)=z(1-tz^2)(1-z+tz-z^2-tz^2-t^2*z^3)/[(1-z-tz^2)(1-z-z^2-3tz^2-tz^3+t^2*z^4)].
%e A121529 T(4,2)=2 because we have U/UDDU/UDD and U/UU/UDDDD, where U=(1,1) and 
               D=(1,-1) (the double rises at an odd level are indicated by a /).
%e A121529 Triangle starts:
%e A121529 1;
%e A121529 1,1;
%e A121529 1,4;
%e A121529 1,10,2;
%e A121529 1,19,14;
%e A121529 1,33,50,5;
%p A121529 G:=z*(1-t*z^2)*(1-z+t*z-z^2-t*z^2-t^2*z^3)/(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..floor(n/
               2)) od; # yields sequence in triangular form
%Y A121529 Cf. A001519, A121530, A121531, A054142.
%Y A121529 Sequence in context: A064947 A059926 A138775 this_sequence A006370 A108759 
               A158824
%Y A121529 Adjacent sequences: A121526 A121527 A121528 this_sequence A121530 A121531 
               A121532
%K A121529 nonn,tabf
%O A121529 1,5
%A A121529 Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 05 2006

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