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A166295 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k UUDUDD's starting at level 0 (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).

%I A166295 #7 Jul 21 2017 02:47:41
%S A166295 1,1,2,3,1,6,2,12,5,26,10,1,57,22,3,128,48,9,291,109,22,1,670,250,54,
%T A166295 4,1558,582,129,14,3655,1366,311,40,1,8639,3232,750,109,5,20554,7696,
%U A166295 1818,284,20,49185,18432,4419,730,65,1,118301,44368,10776,1856,195,6
%N A166295 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k UUDUDD's starting at level 0 (0 <= k <= floor(n/3); U=(1,1), D=(1,-1)).
%C A166295 Row n has 1 + floor(n/3) terms.
%C A166295 Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
%C A166295 T(n,0) = A166296(n).
%C A166295 Sum_{k=0..floor(n/3)} k*T(n,k) = A166297(n).
%F A166295 G.f.: G(t,z) = 1/(1-z-z^2+z^3-t*z^3-z^3*g), where g = 1+zg + z^2*g + z^3*g^2.
%e A166295 T(4,1)=2 because we have UDUUDUDD and UUDUDDUD.
%e A166295 Triangle starts:
%e A166295    1;
%e A166295    1;
%e A166295    2;
%e A166295    3,  1;
%e A166295    6,  2;
%e A166295   12,  5;
%e A166295   26, 10,  1;
%p A166295 G := 2/(1-z-z^2+2*z^3-2*t*z^3+sqrt(1-2*z-z^2-2*z^3+z^4)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form
%Y A166295 Cf. A004148, A166296, A166297.
%K A166295 nonn,tabf
%O A166295 0,3
%A A166295 _Emeric Deutsch_, Oct 29 2009