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Search: id:A118920
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| A118920 |
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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross the x-axis k times (n>=1, k>=0).(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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2
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| 2, 4, 2, 10, 8, 2, 28, 28, 12, 2, 84, 96, 54, 16, 2, 264, 330, 220, 88, 20, 2, 858, 1144, 858, 416, 130, 24, 2, 2860, 4004, 3276, 1820, 700, 180, 28, 2, 9724, 14144, 12376, 7616, 3400, 1088, 238, 32, 2, 33592, 50388, 46512, 31008, 15504, 5814, 1596, 304, 36, 2
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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Row sums are the central binomial coefficients (A000984). T(n,0)=2*A000108(n) (the Catalan numbers doubled). T(n,1)=2*A002057(n-2). Sum(k*T(n,k),k>=0)=2*A008549(n-1). For crossings of the x-axis in one direction, see A118919.
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FORMULA
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T(n,k)=2(k+1)binomial(2n,n-k-1)/n. G.f.=G(t,z)=2zC^2/(1-tzC^2), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. More generally, the trivariate g.f. G=G(x,y,z), where x (y) marks number of downward (upward) crossings of the x-axis, is given by G=zC^2*[2+(x+y)zC^2]/(1-xyz^2*C^4).
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EXAMPLE
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T(3,1)=8 because we have ud|dudu,ud|dduu,udud|du,uudd|du,du|udud,du|uudd, dudu|ud, and dduu|ud (the crossings of the x-axis are shown by |).
Triangle starts:
2;
4,2;
10,8,2;
28,28,12,2;
84,96,54,16,2;
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MAPLE
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T:=(n, k)->2*(k+1)*binomial(2*n, n-k-1)/n: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000984, A000108, A002057, A008549, A118919.
Sequence in context: A099585 A097577 A097692 this_sequence A125755 A118921 A121799
Adjacent sequences: A118917 A118918 A118919 this_sequence A118921 A118922 A118923
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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), May 06 2006
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