A166291 - OEIS

%I #2 Mar 30 2012 17:36:23

%S 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 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 214,240,200,98,56,8,9,0,1,55,210,421,549,481,335,140,72,9,10,0,1,89,370

%N 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 Sum of entries in row n = A004148(n+1) (the secondary structure numbers).

%C T(n,0)=A000045(n-1) (the Fibonacci numbers).

%C Sum(k*T(n,k), k=0..n)=A166292(n).

%F 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 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 Triangle starts:

%e 1;

%e 0,1;

%e 1,0,1;

%e 1,2,0,1;

%e 2,2,3,0,1;

%e 3,6,3,4,0,1.

%p 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 Cf. A004148, A000045, A166292, A166293, A166294

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Oct 12 2009