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A166285 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n having k peak plateaux (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

%I A166285 #12 Oct 08 2024 06:25:24
%S A166285 1,1,2,3,1,5,3,9,8,17,19,1,34,43,5,70,97,18,147,219,56,1,313,498,160,
%T A166285 7,673,1140,438,32,1459,2623,1168,122,1,3185,6061,3062,418,9,6995,
%U A166285 14053,7932,1342,50,15445,32677,20360,4124,225,1,34265,76171,51886,12274,895
%N A166285 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n having k peak plateaux (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).
%C A166285 A peak plateau is a run of consecutive peaks that is preceded by an upstep U and followed by a down step D; a peak consists of an upstep followed by a downstep.
%C A166285 Row n contains 1 + floor(n/3) entries.
%C A166285 Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
%C A166285 T(n,0) = A166286(n).
%C A166285 Sum_{k=0..floor(n/3)} k*T(n,k) = A166287(n).
%F A166285 G.f. G=G(t,z) satisfies G = 1 + zG + z^2*G + z^3*G[G+(t-1)/(1-z)].
%e A166285 T(4,1)=3 because we have UD(UUDUDD), (UUDUDD)UD, and (UUDUDUDD) (the peak plateaux are shown between parentheses).
%e A166285 Triangle starts:
%e A166285    1;
%e A166285    1;
%e A166285    2;
%e A166285    3,  1;
%e A166285    5,  3;
%e A166285    9,  8;
%e A166285   17, 19,  1;
%e A166285   34, 43,  5;
%p A166285 F := RootOf(G = 1+z*G+z^2*G+z^3*G*((t-1)/(1-z)+G), G): Fser := series(F, z = 0, 20): for n from 0 to 17 do P[n] := sort(coeff(Fser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
%Y A166285 Cf. A004148, A166286, A166287.
%K A166285 nonn,tabf
%O A166285 0,3
%A A166285 _Emeric Deutsch_, Oct 12 2009