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Search: id:A166299
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| A166299 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having k UUDD's starting at level 0. |
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3
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| 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 2, 0, 5, 2, 0, 1, 10, 4, 3, 0, 22, 11, 3, 0, 1, 50, 22, 6, 4, 0, 113, 49, 18, 4, 0, 1, 260, 114, 36, 8, 5, 0, 605, 260, 81, 26, 5, 0, 1, 1418, 604, 193, 52, 10, 6, 0, 3350, 1419, 444, 118, 35, 6, 0, 1, 7967, 3350, 1041, 288, 70, 12, 7, 0, 19055, 7966
(list; graph; listen)
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OFFSET
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0,10
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COMMENT
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Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).
Number of entries in row n is 1 + floor(n/2).
T(n,0)=A166300(n).
Sum(k*T(n,k), k>=0)=A075125(n+2).
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FORMULA
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G.f.=G(t,z)=1/(1 + z - zg - tz^2), where g=g(z) satisfies g=1 + zg(g - 1 + z).
G.f. of column k is z^{2k}/(1 + z - zg)^{k+1} (k>=0).
G(t,z)=2/[1+z+z^2+sqrt((1+z+z^2)(1-3z+z^2)-2tz^2)].
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EXAMPLE
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T(7,2)=3 because we have (UUDD)(UUDD)UUUDDD, (UUDD)UUUDDD(UUDD), and UUUDDD(UUDD)(UUDD) (the UUDD's starting at level 0 are shown between parentheses).
Triangle starts:
1;
0;
0,1;
1,0;
1,0,1;
2,2,0;
5,2,0,1;
10,4,3,0;
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MAPLE
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G := 2/(1+z+z^2+sqrt((1+z+z^2)*(1-3*z+z^2))-2*t*z^2): Gser := simplify(series(G, z = 0, 18)): 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, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A004148, A166300, A075125
Sequence in context: A160125 A151868 A052079 this_sequence A088972 A168505 A100334
Adjacent sequences: A166296 A166297 A166298 this_sequence A166300 A166301 A166302
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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), Nov 07 2009
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