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Search: id:A166288
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| A166288 |
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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 and having k UDUD's (0<=k <= n-1; U=(1,1), D=(1,-1)). |
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+0
3
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| 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 4, 5, 6, 1, 1, 6, 12, 9, 8, 1, 1, 9, 23, 24, 14, 10, 1, 1, 17, 38, 56, 40, 20, 12, 1, 1, 26, 84, 100, 110, 60, 27, 14, 1, 1, 46, 145, 250, 210, 190, 84, 35, 16, 1, 1, 81, 280, 480, 580, 385, 301, 112, 44, 18, 1, 1, 135, 551, 995, 1225, 1155, 644, 448
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0)=A166289(n).
Sum(k*T(n,k), k=0..n-1)=A166290(n).
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FORMULA
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G.f. = G(t,z) -1, where G=G(t,z) satisfies z^3*G^2 - (1+z-tz)(1-tz-z^2)G+(1+z-tz)^2=0.
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EXAMPLE
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T(5,2)=6 because we have (UDUDUD)UUDD, UDU(UDUDUD)D, UUDD(UDUDUD), U(UDUD)D(UDUD), U(UDUDUD)DUD, and (UDUD)U(UDUD)D (the UDUD's are shown between parentheses).
Triangle starts:
1;
1,1;
2,1,1;
2,4,1,1;
4,5,6,1,1;
6,12,9,8,1,1;
9,23,24,14,10,1,1;
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MAPLE
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F := RootOf(z^3*G^2-(1+z-t*z)*(1-t*z-z^2)*G+(1+z-t*z)^2, G): Fser := series(F, z = 0, 15): for n to 12 do P[n] := sort(coeff(Fser, z, n)) end do: for n to 12 do seq(coeff(P[n], t, j), j = 0 .. n-1) end do; # yields sequence in triangular form
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CROSSREFS
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A004148, A166289, A166290
Sequence in context: A136788 A136450 A131054 this_sequence A056648 A056061 A029265
Adjacent sequences: A166285 A166286 A166287 this_sequence A166289 A166290 A166291
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 12 2009
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