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A101475
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.
1
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
OFFSET
0,3
LINKS
W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
FORMULA
T(n, k) = Sum_{i>=0} (binomial(2n, n-i*(k+2)) - binomial(2n, n+i*(k+2)+k+1)).
EXAMPLE
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;
MATHEMATICA
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 *)
CROSSREFS
Left-hand columns include A001700 and A024718. Right-hand columns include A000984 and A030662. Row sums are in A101476.
Sequence in context: A347868 A039848 A018494 * A262931 A018524 A057035
KEYWORD
nonn,tabl,walk
AUTHOR
Ralf Stephan, Jan 21 2005
STATUS
approved