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A101475
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Triangle T(n,k) read by rows: number of lattice paths from (0,0) to (0,2n) with steps (1,1) or (1,-1) that stay between the lines y=0 and y=k.
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1
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1, 1, 2, 3, 5, 6, 10, 15, 19, 20, 35, 50, 63, 69, 70, 126, 176, 217, 243, 251, 252, 462, 638, 770, 870, 913, 923, 924, 1716, 2354, 2794, 3159, 3355, 3419, 3431, 3432, 6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870, 24310, 33099, 38489
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OFFSET
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0,3
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LINKS
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FORMULA
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T(n, k) = Sum_{i>=0} (binomial(2n, n-i*(k+2)) - binomial(2n, n+i*(k+2)+k+1)).
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EXAMPLE
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Triangle begins
1;
1, 2;
3, 5, 6;
10, 15, 19, 20;
35, 50, 63, 69, 70;
126, 176, 217, 243, 251, 252;
462, 638, 770, 870, 913, 923, 924;
1716, 2354, 2794, 3159, 3355, 3419, 3431, 3432;
6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870;
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MATHEMATICA
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T[n_, k_] := Sum[Binomial[2n, n-i(k+2)] - Binomial[2n, n+i(k+2)+k+1], {i, 0, n}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2019 *)
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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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