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Search: id:A114499
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| A114499 |
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Triangle read by rows: number of Dyck paths of semilength n having k 3-bridges of a given shape (0<=k<=floor(n/3)). A 3-bridge is a subpath of the form UUUUDDD or UUDUDD starting at level 0. |
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+0
5
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| 1, 1, 2, 4, 1, 12, 2, 37, 5, 119, 12, 1, 390, 36, 3, 1307, 114, 9, 4460, 376, 25, 1, 15452, 1262, 78, 4, 54207, 4310, 255, 14, 192170, 14934, 863, 44, 1, 687386, 52397, 2967, 145, 5, 2477810, 185780, 10338, 492, 20, 8992007, 664631, 36424, 1712, 70, 1
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Row n has 1+floor(n/3) terms. Row sums are the Catalan numbers (A000108). Column 0 is A114500. Sum(kT(n,k),k=0..floor(n/3))=Catalan(n-2) (n>=3; A000108).
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FORMULA
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G.f.=1/(1+z^3-tz^3-zC), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(4,1)=2 because we have UD(UUUDDD) and (UUUDDD)UD (or UD(UUDUDD) and (UUDUDD)UD). The 3-bridges are shown between parentheses.
Triangle starts:
1;
1;
2;
4,1;
12,2;
37,5;
119,12,1;
390,36,3;
1307,114,9;
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z^3-t*z^3): Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 17 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/3)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A114500.
Adjacent sequences: A114496 A114497 A114498 this_sequence A114500 A114501 A114502
Sequence in context: A048941 A135333 A124503 this_sequence A030730 A117131 A030043
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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 04 2005
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