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%I #13 Nov 26 2019 04:31:10
%S 1,1,1,1,3,1,1,6,5,1,1,9,15,8,1,1,12,34,30,11,1,1,15,62,85,55,14,1,1,
%T 18,99,200,185,89,17,1,1,21,145,402,510,365,132,20,1,1,24,200,718,
%U 1220,1160,650,184,23,1,1,27,264,1175,2585,3155,2400,1067,245,26,1,1,30,337
%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 odd level (0 <= k <= n-1).
%C 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-indexed Fibonacci numbers (A001519).
%C T(n,k) = A121522(n,n-k), i.e., triangle is mirror image of A121522.
%C Sum_{k>=0} k*T(n,k) = A121525(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-tz^2)(1-2tz^2-t^2*z^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 UDU(U)D(U)DD, U(U)DDU(U)DD, U(U)D(U)UDDD, U(U)UDD(U)DD and U(U)U(U)DDDD, where U=(1,1) and D=(1,-1) (the up steps starting at an odd 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, 6, 5, 1;
%e 1, 9, 15, 8, 1;
%e 1, 12, 34, 30, 11, 1;
%p g:=z*(1-t*z^2)*(1-2*t*z^2-t^2*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=0..n-1) od; # yields sequence in triangular form
%t G[t_, z_] = z*(1 - t*z^2)*(1 - 2*t*z^2 - t^2*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);
%t T[n_, k_] := SeriesCoefficient[G[t, z], {z, 0, n}, {t, 0, k}];
%t Table[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* _Jean-François Alcover_, Jan 15 2018 *)
%Y Cf. A001519, A121522, A121525.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Aug 05 2006
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