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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.

%I A101475 #14 Jan 20 2019 23:20:14
%S A101475 1,1,2,3,5,6,10,15,19,20,35,50,63,69,70,126,176,217,243,251,252,462,
%T A101475 638,770,870,913,923,924,1716,2354,2794,3159,3355,3419,3431,3432,6435,
%U A101475 8789,10307,11610,12430,12766,12855,12869,12870,24310,33099,38489
%N 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.
%H A101475 W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, <a href="https://arxiv.org/abs/math/0501230">Crossings and nestings of matchings and partitions</a>, arXiv:math/0501230 [math.CO], 2005.
%F A101475 T(n, k) = Sum_{i>=0} (binomial(2n, n-i*(k+2)) - binomial(2n, n+i*(k+2)+k+1)).
%e A101475 Triangle begins
%e A101475      1;
%e A101475      1,    2;
%e A101475      3,    5,     6;
%e A101475     10,   15,    19,    20;
%e A101475     35,   50,    63,    69,    70;
%e A101475    126,  176,   217,   243,   251,   252;
%e A101475    462,  638,   770,   870,   913,   923,   924;
%e A101475   1716, 2354,  2794,  3159,  3355,  3419,  3431,  3432;
%e A101475   6435, 8789, 10307, 11610, 12430, 12766, 12855, 12869, 12870;
%t A101475 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 *)
%Y A101475 Left-hand columns include A001700 and A024718. Right-hand columns include A000984 and A030662. Row sums are in A101476.
%K A101475 nonn,tabl,walk
%O A101475 0,3
%A A101475 _Ralf Stephan_, Jan 21 2005