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Search: id:A118919
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| A118919 |
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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). |
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+0
5
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| 1, 2, 5, 1, 14, 6, 42, 27, 1, 132, 110, 10, 429, 429, 65, 1, 1430, 1638, 350, 14, 4862, 6188, 1700, 119, 1, 16796, 23256, 7752, 798, 18, 58786, 87210, 33915, 4655, 189, 1, 208012, 326876, 144210, 24794, 1518, 22, 742900, 1225785, 600875, 123970, 10350
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920.
Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - Paul D. Hanna (pauldhanna(AT)juno.com), May 10 2006
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FORMULA
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T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud, and duud\du (the downward crossings of the x-axis are shown by a back-slash \).
Triangle starts:
1;
2;
5,1;
14,6;
42,27,1;
132,110,10;
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MAPLE
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T:=(n, k)->(2*k+1)*binomial(2*n+2, n-2*k)/(n+1): for n from 0 to 13 do seq(T(n, k), k=0..floor(n/2)) od; # yields sequence in triangular form
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PROGRAM
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(PARI) T(n, k)=if(n<2*k|k<0, 0, (2*k+1)*binomial(2*n+2, n-2*k)/(n+1)) - Paul D. Hanna (pauldhanna(AT)juno.com), May 10 2006
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CROSSREFS
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Cf. A000984, A000108, A003517, A003519, A008549, A118920.
Cf. A119243 (eigenvector), A119245 (variant).
Sequence in context: A118964 A073187 A138159 this_sequence A101282 A101895 A110220
Adjacent sequences: A118916 A118917 A118918 this_sequence A118920 A118921 A118922
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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), May 06 2006
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