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A132890
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Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have height k (1<=k<=n).
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1
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1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 1, 7, 5, 5, 1, 1, 1, 7, 13, 6, 6, 1, 1, 1, 15, 18, 20, 7, 7, 1, 1, 1, 15, 39, 26, 27, 8, 8, 1, 1, 1, 31, 57, 73, 35, 35, 9, 9, 1, 1, 1, 31, 112, 99, 109, 44, 44, 10, 10, 1, 1, 1, 63, 169, 253, 152, 154, 54, 54, 11, 11, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Sum of terms in row n = binom(n, floor(n/2))=A001405(n). T(n,2)=A052551(n-2) (n>=2). T(n,3)=A005672(n)=Fibonacci(n+1)-2^floor(n/2). Sum(k*T(n,k),k=1..n)=A132891(n).
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REFERENCES
| R. Kemp, On the average depth of a prefix of the Dycklanguage D_1, Discrete Math., 36, 1981, 155-170.
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FORMULA
| The g.f. of column k is g(k, z)=v^k*(1+v)*(1+v^2)*/((1+v^(k+1))*(1+v^(k+2))), where v = (1-sqrt(1-4*z^2))/(2*z). (Obtained as the difference G(k,z)-G(k-1,z), where G(k,z) is given in the R. Kemp reference (p. 159)).
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EXAMPLE
| T(5,3)=4 because we have UDUUU, UUDUU, UUUDD and UUUDU, where U=(1,1) and D=(1,-1).
Triangle starts:
1;
1,1;
1,1,1;
1,3,1,1;
1,3,4,1;1;
1,7,5,5,1,1;
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MAPLE
| v := ((1-sqrt(1-4*z^2))*1/2)/z: g := proc (k) options operator, arrow: v^k*(1+v)*(1+v^2)/((1+v^(k+1))*(1+v^(k+2))) end proc: T := proc (n, k) options operator, arrow: coeff(series(g(k), z = 0, 50), z, n) end proc: for n from 0 to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form
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CROSSREFS
| Cf. A001405, A052551, A005672, A132891, A068914
Sequence in context: A180683 A166030 A191523 * A069290 A076476 A016733
Adjacent sequences: A132887 A132888 A132889 * A132891 A132892 A132893
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 08 2007
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