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%I #24 Sep 14 2021 10:50:23
%S 1,1,1,1,1,2,1,1,2,5,1,1,2,5,14,1,1,2,5,15,42,1,1,2,5,15,51,132,1,1,2,
%T 5,15,52,188,429,1,1,2,5,15,52,202,731,1430,1,1,2,5,15,52,203,856,
%U 2950,4862,1,1,2,5,15,52,203,876,3868,12235,16796
%N Array giving number of (k,2)-noncrossing partitions of [n], read by antidiagonals.
%C A partition is (k,2)-noncrossing if it avoids the pattern 12...k12.
%H Toufik Mansour and Simone Severini, <a href="https://doi.org/10.1016/j.disc.2007.08.068">Enumeration of (k,2)-noncrossing partitions</a>, Discrete Math., 308 (2008), 4570-4577.
%e Table begins:
%e k\n| 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 2| 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012
%e 3| 1 1 2 5 15 51 188 731 2950 12235 51822 223191 974427
%e 4| 1 1 2 5 15 52 202 856 3868 18313 89711 450825 2310453
%e 5| 1 1 2 5 15 52 203 876 4112 20679 109853 608996 3488806
%e 6| 1 1 2 5 15 52 203 877 4139 21111 115219 666388 4045991
%t b[j_, j_] := 1;
%t b[i_, j_] := j x Product[s x - 1, {s, i + 1, j - 1}];
%t y[k_] := (1 - (k - 2) x - Sqrt[(1 - k x)^2 - 4 x^2]) / (2 x (1 - (k - 2) x));
%t s[k_, op_] := Sum[(-1)^(i + j) op[x, i] b[i, j], {j, 0, k - 2}, {i, 0, j}];
%t p[k_] := (x^(k - 1) y[k]/(1 - x y[k]) + s[k, Power]) / (1 - s[k, Times]);
%t t[n_, k_] := SeriesCoefficient[p[k], {x, 0, n}];
%t Print@Flatten@Table[t[n, ad - n + 2], {ad, 0, 10}, {n, 0, ad}]
%t (* _Andrey Zabolotskiy_, Sep 14 2021 *)
%Y Rows include A000108, A007317, A140980, A141080, A141081.
%Y Cf. A000110.
%K nonn,tabl
%O 0,6
%A _Jonathan Vos Post_, Dec 10 2006
%E Offset corrected by _Joerg Arndt_, Apr 18 2014
%E More terms from _Andrey Zabolotskiy_, Sep 14 2021
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