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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)).
2
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, 7, 673, 1140, 438, 32, 1459, 2623, 1168, 122, 1, 3185, 6061, 3062, 418, 9, 6995, 14053, 7932, 1342, 50, 15445, 32677, 20360, 4124, 225, 1, 34265, 76171, 51886, 12274, 895
OFFSET
0,3
COMMENTS
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.
Row n contains 1 + floor(n/3) entries.
Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0) = A166286(n).
Sum_{k=0..floor(n/3)} k*T(n,k) = A166287(n).
FORMULA
G.f. G=G(t,z) satisfies G = 1 + zG + z^2*G + z^3*G[G+(t-1)/(1-z)].
EXAMPLE
T(4,1)=3 because we have UD(UUDUDD), (UUDUDD)UD, and (UUDUDUDD) (the peak plateaux are shown between parentheses).
Triangle starts:
1;
1;
2;
3, 1;
5, 3;
9, 8;
17, 19, 1;
34, 43, 5;
MAPLE
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 12 2009
STATUS
approved