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A256161 Triangle of allowable Stirling numbers of the second kind a(n,k).

%I A256161 #28 Feb 10 2020 10:41:42
%S A256161 1,1,1,1,2,1,1,3,4,1,1,4,11,6,1,1,5,26,23,9,1,1,6,57,72,50,12,1,1,7,
%T A256161 120,201,222,86,16,1,1,8,247,522,867,480,150,20,1,1,9,502,1291,3123,
%U A256161 2307,1080,230,25,1,1,10,1013,3084,10660,10044,6627,2000,355,30,1
%N A256161 Triangle of allowable Stirling numbers of the second kind a(n,k).
%C A256161 Row sums = A007476 starting (1, 2, 4, 9, 23, 65, 199, 654, 2296, 8569, ...).
%C A256161 a(n,k) counts restricted growth words of length n in the letters {1, ..., k} where every even entry appears exactly once.
%H A256161 Yue Cai and Margaret Readdy, <a href="http://arxiv.org/abs/1506.03249">Negative q-Stirling numbers</a>, arXiv:1506.03249 [math.CO], 2015.
%F A256161 a(n,k) = a(n-1,k-1) + ceiling(k/2)*a(n-1,k) for n >= 1 and 1 <= k <= n with boundary conditions a(n,0) = KroneckerDelta[n,0].
%F A256161 a(n,2) = n-1.
%F A256161 a(n,n-1) = floor(n/2)*ceiling(n/2).
%e A256161 a(4,1) = 1 via 1111;
%e A256161 a(4,2) = 3 via 1211, 1121, 1112;
%e A256161 a(4,3) = 4 via 1213, 1231, 1233, 1123;
%e A256161 a(4,4) = 1 via 1234.
%e A256161 Triangle starts:
%e A256161   1;
%e A256161   1,  1;
%e A256161   1,  2,  1;
%e A256161   1,  3,  4,  1;
%e A256161   1,  4, 11,  6,  1;
%e A256161   ...
%t A256161 a[_, 1] = a[n_, n_] = 1;
%t A256161 a[n_, k_] := a[n, k] = a[n-1, k-1] + Ceiling[k/2] a[n-1, k];
%t A256161 Table[a[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 15 2018 *)
%Y A256161 Cf. A007476 (row sums), A246118 (essentially the same triangle).
%K A256161 nonn,tabl
%O A256161 1,5
%A A256161 _Margaret A. Readdy_, Mar 16 2015