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Search: id:A114655
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| A114655 |
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Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps. |
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4
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| 2, 4, 2, 8, 12, 2, 16, 48, 24, 2, 32, 160, 160, 40, 2, 64, 480, 800, 400, 60, 2, 128, 1344, 3360, 2800, 840, 84, 2, 256, 3584, 12544, 15680, 7840, 1568, 112, 2, 512, 9216, 43008, 75264, 56448, 18816, 2688, 144, 2, 1024, 23040, 138240, 322560, 338688, 169344
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OFFSET
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1,1
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COMMENT
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Row sums are the large Schroeder numbers (A006318). Sum(k*T(n,k),k=1..n)=A002003(n). T(n,k)=2*A114656(n,k).
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FORMULA
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T(n, k)=2^(n-k+1)*binomial(n, k)*binomial(n, k-1)/n (1<=k<=n). G.f. G=G(t, z) satisfies G=z(2+G)(t+G).
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EXAMPLE
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T(3,3)=2 because we have (U)D(U)D(H) and (U)D(U)D(U)D, where U=(1,1), D=(1,-1) and H=(2,0) (the weak ascents are shown between parentheses).
Triangle starts:
2;
4,2;
8,12,2;
16,48,24,2;
32,160,160,40,2.
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MAPLE
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T:=(n, k)->2^(n-k+1)*binomial(n, k)*binomial(n, k-1)/n: 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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Cf. A006318, A002003, A114656.
Sequence in context: A065286 A068217 A114593 this_sequence A051288 A120434 A008303
Adjacent sequences: A114652 A114653 A114654 this_sequence A114656 A114657 A114658
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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 23 2005
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