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Search: id:A121522
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| A121522 |
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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 even level (1<=k<=n). A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing. |
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+0
3
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| 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 8, 15, 9, 1, 1, 11, 30, 34, 12, 1, 1, 14, 55, 85, 62, 15, 1, 1, 17, 89, 185, 200, 99, 18, 1, 1, 20, 132, 365, 510, 402, 145, 21, 1, 1, 23, 184, 650, 1160, 1220, 718, 200, 24, 1, 1, 26, 245, 1067, 2400, 3155, 2585, 1175, 264, 27, 1, 1, 29, 315
(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)=A121524(n,n-k), i.e. triangle is mirror image of A121524. Sum(k*T(n,k), k=1..n)=A121523(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)=tz(1-tz^2)(1-2tz^2-tz^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 (U)D(U)UDUDD, (U)UDD(U)UDD, (U)UDU(U)DDD, (U)U(U)DDUDD and (U)U(U)UDDDD, where U=(1,1) and D=(1,-1) (the up steps starting at an even level are shown between parentheses; UUDUDDUD does not qualify because it is not nondecreasing).
Triangle starts:
1;
1,1;
1,3,1;
1,5,6,1;
1,8,15,9,1;
1,11,30,34,12,1;
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MAPLE
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g:=t*z*(1-t*z^2)*(1-2*t*z^2-t*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=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001519, A121523, A121524.
Sequence in context: A076756 A054142 A114172 this_sequence A080842 A145661 A119258
Adjacent sequences: A121519 A121520 A121521 this_sequence A121523 A121524 A121525
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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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