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A082680
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Triangle read by rows: T(n,k) is the number of 2-stack sortable n-permutations with k runs.
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8
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1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 49, 20, 1, 1, 35, 168, 168, 35, 1, 1, 56, 462, 900, 462, 56, 1, 1, 84, 1092, 3630, 3630, 1092, 84, 1, 1, 120, 2310, 12012, 20449, 12012, 2310, 120, 1, 1, 165, 4488, 34320, 91091, 91091, 34320, 4488, 165, 1, 1, 220, 8151, 87516, 340340, 529984, 340340, 87516, 8151, 220, 1
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OFFSET
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1,5
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COMMENTS
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Number of beta(1,0)-trees on n+1 nodes with k leaves.
T(n,k) is the number of rooted non-separable planar maps with n+1 edges, k+1 faces and n+2-k vertices. - Andrew Howroyd, Mar 29 2021
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LINKS
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FORMULA
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T(n, k) = (n+k-1)!*(2*n-k)!/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!).
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EXAMPLE
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Triangle starts:
1;
1, 1;
1, 4, 1;
1, 10, 10, 1;
1, 20, 49, 20, 1;
1, 35, 168, 168, 35, 1;
1, 56, 462, 900, 462, 56, 1;
1, 84, 1092, 3630, 3630, 1092, 84, 1;
...
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MATHEMATICA
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Table[(n+k-1)!(2n-k)!/k!/(n+1-k)!/(2k-1)!/(2n-2k+1)!, {n, 10}, {k, n}]//Flatten (* Harvey P. Dale, Jun 10 2020 *)
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PROG
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(PARI) T(n, k) = (n+k-1)!*(2*n-k)!/k!/(n+1-k)!/(2*k-1)!/(2*n-2*k+1)! \\ Andrew Howroyd, Mar 29 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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