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A166291 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 peaks at odd level (0<=k<=n; U=(1,1), D=(1,-1)).

%I A166291 #2 Mar 30 2012 17:36:23
%S A166291 1,0,1,1,0,1,1,2,0,1,2,2,3,0,1,3,6,3,4,0,1,5,10,12,4,5,0,1,8,20,22,20,
%T A166291 5,6,0,1,13,36,52,40,30,6,7,0,1,21,66,104,109,65,42,7,8,0,1,34,118,
%U A166291 214,240,200,98,56,8,9,0,1,55,210,421,549,481,335,140,72,9,10,0,1,89,370
%N A166291 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 peaks at odd level (0<=k<=n; U=(1,1), D=(1,-1)).
%C A166291 Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
%C A166291 T(n,0)=A000045(n-1) (the Fibonacci numbers).
%C A166291 Sum(k*T(n,k), k=0..n)=A166292(n).
%F A166291 The trivariate g.f. G=G(t,s,z), where z marks semilength, t marks odd-level peaks and s marks even-level peaks, satisfies G = 1 + tzG + sz^2*G + s^2*z^3*HG, where H=G(s,t,z) (interchanging t and s and eliminating H, one obtains G(t,s,z); see the Maple program).
%e A166291 T(4,2)=3 because we have (UD)(UD)UUDD, (UD)UUDD(UD), and UUDD(UD)(UD) (the odd level peaks are shown between parentheses).
%e A166291 Triangle starts:
%e A166291 1;
%e A166291 0,1;
%e A166291 1,0,1;
%e A166291 1,2,0,1;
%e A166291 2,2,3,0,1;
%e A166291 3,6,3,4,0,1.
%p A166291 p1 := -G+1+t*z*G+s*z^2*G+s^2*z^3*H*G: p2 := subs({t = s, s = t, G = H, H = G}, p1): r := resultant(p1, p2, H): G := RootOf(subs(s = 1, r), G): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; yields sequence in triangular form
%Y A166291 Cf. A004148, A000045, A166292, A166293, A166294
%K A166291 nonn,tabl
%O A166291 0,8
%A A166291 _Emeric Deutsch_, Oct 12 2009