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A121529
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Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k double rises at an odd level (n>=1, k>=0). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing.
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2
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1, 1, 1, 1, 4, 1, 10, 2, 1, 19, 14, 1, 33, 50, 5, 1, 55, 132, 45, 1, 90, 301, 205, 13, 1, 146, 631, 680, 139, 1, 236, 1255, 1892, 763, 34, 1, 381, 2409, 4717, 3019, 419, 1, 615, 4509, 10920, 9846, 2677, 89, 1, 993, 8283, 23974, 28292, 12241, 1241, 1, 1604, 14998
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OFFSET
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1,5
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COMMENTS
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Row n contains 1+floor(n/2) terms. Row sums are the odd-subscripted Fibonacci numbers (A001519). T(2n,n)=Fibonacci(2n-1) (A001519). Sum(k*T(n,k), k>=0)=A121530(n).
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LINKS
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Table of n, a(n) for n=1..58.
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
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FORMULA
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G.f.: G(t,z)=z(1-tz^2)(1-z+tz-z^2-tz^2-t^2*z^3)/[(1-z-tz^2)(1-z-z^2-3tz^2-tz^3+t^2*z^4)].
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EXAMPLE
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T(4,2)=2 because we have U/UDDU/UDD and U/UU/UDDDD, where U=(1,1) and D=(1,-1) (the double rises at an odd level are indicated by a /).
Triangle starts:
1;
1,1;
1,4;
1,10,2;
1,19,14;
1,33,50,5;
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MAPLE
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G:=z*(1-t*z^2)*(1-z+t*z-z^2-t*z^2-t^2*z^3)/(1-z-t*z^2)/(1-z-z^2-3*t*z^2-t*z^3+t^2*z^4): Gser:=simplify(series(G, z=0, 18)): for n from 1 to 15 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 15 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001519, A121530, A121531, A054142.
Sequence in context: A059926 A138775 A209385 * A006370 A262370 A108759
Adjacent sequences: A121526 A121527 A121528 * A121530 A121531 A121532
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Aug 05 2006
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STATUS
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approved
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