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Search: id:A121524
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| A121524 |
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Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n and having k up steps starting at an odd level (0<=k<=n-1). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing. |
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3
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| 1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 9, 15, 8, 1, 1, 12, 34, 30, 11, 1, 1, 15, 62, 85, 55, 14, 1, 1, 18, 99, 200, 185, 89, 17, 1, 1, 21, 145, 402, 510, 365, 132, 20, 1, 1, 24, 200, 718, 1220, 1160, 650, 184, 23, 1, 1, 27, 264, 1175, 2585, 3155, 2400, 1067, 245, 26, 1, 1, 30, 337
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,k)=A121522(n,n-k), i.e. triangle is mirror image of A121522. Sum(k*T(n,k), k=0..n-1)=A121525(n).
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REFERENCES
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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)(1-2tz^2-t^2*z^3)/(1-z-tz-4tz^2+2tz^3+2t^2*z^3+6t^2*z^4-t^3*z^6).
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EXAMPLE
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T(4,2)=5 because we have UDU(U)D(U)DD, U(U)DDU(U)DD, U(U)D(U)UDDD, U(U)UDD(U)DD and U(U)U(U)DDDD, where U=(1,1) and D=(1,-1) (the up steps starting at an odd level are shown between parentheses; UUDUDDUD does not qualify because it is not nondecreasing).
Triangle starts:
1;
1,1;
1,3,1;
1,6,5,1;
1,9,15,8,1;
1,12,34,30,11,1;
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MAPLE
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g:=z*(1-t*z^2)*(1-2*t*z^2-t^2*z^3)/(1-z-t*z-4*t*z^2+2*t*z^3+2*t^2*z^3+6*t^2*z^4-\ t^3*z^6): gser:=simplify(series(g, z=0, 17)): for n from 1 to 12 do P[n]:=sort(expand(coeff(gser, z, n))) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001519, A121522, A121525.
Sequence in context: A128101 A124802 A102036 this_sequence A103141 A085478 A123970
Adjacent sequences: A121521 A121522 A121523 this_sequence A121525 A121526 A121527
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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), Aug 05 2006
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