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Search: id:A114486
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| A114486 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k UUDD's starting at level 0; here U=(1,1), D=(1,-1) (0<=k<=floor(n/2)). |
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+0
2
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| 1, 1, 1, 1, 3, 2, 10, 3, 1, 31, 8, 3, 98, 27, 6, 1, 321, 88, 16, 4, 1078, 287, 54, 10, 1, 3686, 960, 183, 28, 5, 12789, 3280, 616, 95, 15, 1, 44919, 11378, 2106, 332, 45, 6, 159407, 39953, 7323, 1152, 155, 21, 1, 570704, 141752, 25785, 4028, 556, 68, 7, 2058817
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OFFSET
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0,5
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COMMENT
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Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(k*T(n,k),k=0..floor(n/2))=A000108(n-1) (the Catalan numbers). Column 0 yields A114487.
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REFERENCES
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A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
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FORMULA
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G.f. G=G(t, z) satisfies G=1+z(C-z+tz)G, where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. G=2/[1+2z^2-2tz^2+sqrt(1-4z)].
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EXAMPLE
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T(5,2)=3 because we have UUDDUUDDUD, UUDDUDUUDD and UDUUDDUUDD, where U=(1,1), D=(1,-1).
Triangle starts:
1;
1;
1,1;
3,2;
10,3,1;
31,8,3;
98,27,6,1; ...
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: eq:=G=1+z*(C-z+t*z)*G: G:=solve(eq, G): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 15 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 15 do seq(coeff(t*P[n], t^j), j=1..1+floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A114487.
Sequence in context: A049971 A092502 A135515 this_sequence A090780 A063549 A057977
Adjacent sequences: A114483 A114484 A114485 this_sequence A114487 A114488 A114489
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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), Nov 30 2005
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