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Search: id:A166285
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| A166285 |
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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)). 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. |
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+0
3
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| 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
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OFFSET
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0,3
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COMMENT
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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*T(n,k), k=0..floor(n/3))=A166287(n).
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FORMULA
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G.f. G=G(t,z) satisfies G = 1 + zG + z^2*G + z^3*G[G+(t-1)/(1-z)].
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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A004148, A166286, A166287
Sequence in context: A111609 A047706 A114711 this_sequence A002472 A060116 A114690
Adjacent sequences: A166282 A166283 A166284 this_sequence A166286 A166287 A166288
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2009
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