|
%I #5 Mar 06 2018 11:40:03
%S 1,1,1,1,3,1,1,5,6,1,1,8,15,9,1,1,11,30,34,12,1,1,14,55,85,62,15,1,1,
%T 17,89,185,200,99,18,1,1,20,132,365,510,402,145,21,1,1,23,184,650,
%U 1160,1220,718,200,24,1,1,26,245,1067,2400,3155,2585,1175,264,27,1,1,29,315
%N Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k up steps starting at an even level (1<=k<=n). 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,k)=A121524(n,n-k), i.e. triangle is mirror image of A121524. Sum(k*T(n,k), k=1..n)=A121523(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(1-tz^2)(1-2tz^2-tz^3)/(1-z-tz-4tz^2+2tz^3+2t^2*z^3+6t^2*z^4-t^3*z^6).
%e T(4,2)=5 because we have (U)D(U)UDUDD, (U)UDD(U)UDD, (U)UDU(U)DDD, (U)U(U)DDUDD and (U)U(U)UDDDD, where U=(1,1) and D=(1,-1) (the up steps starting at an even level are shown between parentheses; UUDUDDUD does not qualify because it is not nondecreasing).
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,3,1;
%e 1,5,6,1;
%e 1,8,15,9,1;
%e 1,11,30,34,12,1;
%p g:=t*z*(1-t*z^2)*(1-2*t*z^2-t*z^3)/(1-z-t*z-4*t*z^2+2*t*z^3+2*t^2*z^3+6*t^2*z^4-t^3*z^6): gser:=simplify(series(g,z=0,17)): for n from 1 to 12 do P[n]:=sort(expand(coeff(gser,z,n))) od: for n from 1 to 12 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
%Y Cf. A001519, A121523, A121524.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Aug 05 2006
|
