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Search: id:A145895
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| A145895 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having k UDU and DUD's (here U=(1,1), D=(1,-1); 0<=k<=2n-2). |
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1
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| 1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 4, 2, 4, 1, 0, 1, 4, 8, 10, 8, 4, 6, 1, 0, 1, 8, 20, 26, 24, 25, 12, 7, 8, 1, 0, 1, 17, 48, 70, 84, 70, 54, 47, 16, 11, 10, 1, 0, 1, 37, 116, 197, 244, 241, 224, 141, 104, 76, 20, 16, 12, 1, 0, 1, 82, 286, 535, 728, 816, 734, 609, 472, 246, 180, 112, 24, 22
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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Row n contains 2n-1 entries (n>=1).
Row sums are the Catalan numbers (A000108).
T(n,0)=A004148(n-1) (the 2ndary structure numbers).
Sum(k*T(n,k),k=0..2n-2)=2*binom(2n-2,n-2)=2*A001791(n-1).
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FORMULA
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G.f. = c(z/[1+(1-t^2)z+(1-t)^2*z^2]), where c(z)=[1-sqrt(1-4z)](2z) is the g.f. of the Catalan numbers (A000108).
The trivariate g.f., with z marking semilength, t marking number of UDU's and s marking number of DUD's is c(z/[1+(1-ts)z+(1-t)(1-s)z^2]), where c(z)=[1-sqrt(1-4z)](2z) is the g.f. of the Catalan numbers (A000108).
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EXAMPLE
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T(4,3)=4 because we have UDUDUUDD, UUDDUDUD, UDUUDUDD and UUDUDDUD.
Triangle starts:
1;
1;
1,0,1;
1,2,1,0,1;
2,4,2,4,1,0,1;
4,8,10,8,4,6,1,0,1;
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MAPLE
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c := proc (z) options operator, arrow: (1/2-(1/2)*sqrt(1-4*z))/z end proc: G := c(z/(1+(1-t^2)*z+(1-t)^2*z^2)): Gser := simplify(series(G, z = 0, 12)): for n from 0 to 9 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 9 do seq(coeff(P[n], t, k), k = 0 .. 2*n-2) end do; # yields sequence in triangular form
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CROSSREFS
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A000108, A004148, A001791
Sequence in context: A022958 A023444 A136868 this_sequence A114503 A103528 A138352
Adjacent sequences: A145892 A145893 A145894 this_sequence A145896 A145897 A145898
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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), Dec 10 2008
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