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 A121481 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at odd level (0<=k<=n). 3

%I

%S 1,0,1,1,0,1,1,3,0,1,3,3,6,0,1,5,14,5,9,0,1,12,22,35,7,12,0,1,22,68,

%T 53,65,9,15,0,1,49,127,203,97,104,11,18,0,1,94,329,390,444,153,152,13,

%U 21,0,1,201,664,1157,873,816,221,209,15,24,0,1,396,1576,2456,2925,1627,1345

%N Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having k peaks at odd level (0<=k<=n).

%C Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,0)=A121482(n). Sum(k*T(n,k),k=0..n)=A121483(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) = (tz^2+z^2+z-1)(tz^3+2z^2-1)/(1-4z^2-z-tz+2tz^4+4z^4-z^6+2z^3+tz^3+t^2*z^3).

%e T(3,1)=3 because we have U|DUUDUDD, UUDUU|DDD and UUU|DDUDD, where U=(1,1) and D=(1,-1) (the peaks at odd level are shown by a |; the Dyck path UUDUDDUD has 1 peak at odd level but it is not nondecreasing).

%e Triangle starts:

%e 1;

%e 0,1;

%e 1,0,1;

%e 1,3,0,1;

%e 3,3,6,0,1;

%e 5,14,5,9,0,1;

%p G:=(t*z^2+z^2+z-1)*(t*z^3+2*z^2-1)/(1-4*z^2-z-t*z+2*z^4*t+4*z^4-z^6+2*z^3+t*z^3+z^3*t^2): Gser:=simplify(series(G,z=0,15)): P:=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) od; # yields sequence in triangular form

%Y Cf. A001519, A121482, A121483, A121484.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Aug 02 2006

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Last modified March 30 04:12 EDT 2023. Contains 361603 sequences. (Running on oeis4.)