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A118921
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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n having first return to the x-axis at (2k,0) (n,k >= 1). (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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1
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2, 4, 2, 12, 4, 4, 40, 12, 8, 10, 140, 40, 24, 20, 28, 504, 140, 80, 60, 56, 84, 1848, 504, 280, 200, 168, 168, 264, 6864, 1848, 1008, 700, 560, 504, 528, 858, 25740, 6864, 3696, 2520, 1960, 1680, 1584, 1716, 2860, 97240, 25740, 13728, 9240, 7056, 5880, 5280
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OFFSET
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1,1
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COMMENTS
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Row sums are the central binomial coefficients (A000984).
Sum_{k>=1} k*T(n,k) = 2^(2n-1) (A004171).
For returns to the x-axis arriving from above, see A039599.
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LINKS
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FORMULA
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T(n,k) = 2*binomial(2k-2,k-1)*binomial(2n-2k,n-k)/k.
G.f. = G(t,z) = (1-sqrt(1-4tz))/sqrt(1-4z).
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EXAMPLE
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T(3,2)=4 because we have uudd|ud, uudd|du, dduu|ud and dduu|du (first return to the x-axis shown by | ).
Triangle starts:
2;
4, 2;
12, 4, 4;
40, 12, 8, 10;
140, 40, 24, 20, 28;
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MAPLE
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T:=(n, k)->2*binomial(2*k-2, k-1)*binomial(2*n-2*k, n-k)/k: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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