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A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

%I A369324 #16 Jan 21 2024 05:52:04
%S A369324 0,0,1,0,1,1,0,1,2,1,0,1,4,3,1,0,1,8,9,4,1,0,1,16,25,16,5,1,0,1,32,65,
%T A369324 56,25,6,1,0,1,64,161,176,105,36,7,1,0,1,128,385,512,385,176,49,8,1,0,
%U A369324 1,256,897,1408,1281,736,273,64,9,1,0,1,512,2049,3712,3969,2752,1281,400,81,10,1
%N A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.
%H A369324 Toufik Mansour, Howard Skogman, and Rebecca Smith, <a href="https://arxiv.org/abs/2401.06662">Sorting inversion sequences</a>, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.18 at page 10.
%F A369324 A(n,k) = A000035(k) + 2^n*Sum_{i=0..floor((k-2)/2)} binomial(n + k - 3 - 2*i, n - 1).
%F A369324 Sum_{k=0..n} A(n-k,k) = A164039(n-1).
%e A369324 The array begins:
%e A369324   0, 1,  1,   1,   1,    1, ...
%e A369324   0, 1,  2,   3,   4,    5, ...
%e A369324   0, 1,  4,   9,  16,   25, ...
%e A369324   0, 1,  8,  25,  56,  105, ...
%e A369324   0, 1, 16,  65, 176,  385, ...
%e A369324   0, 1, 32, 161, 512, 1281, ...
%e A369324   ...
%t A369324 A[n_,k_]:=(1-(-1)^k)/2+2^n Sum[Binomial[n+k-3-2i,n-1],{i,0,Floor[(k-2)/2]}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten
%Y A369324 Cf. A000004 (k=0), A000012 (k=1), A000079 (k=2), A002064 (k=3), A340257 (k=4).
%Y A369324 Cf. A000290 (n=2), A001477 (n=1), A057427 (n=0), A131423 (n=3), A164039.
%Y A369324 Cf. A000035, A369325 (main diagonal), A369326.
%K A369324 nonn,tabl
%O A369324 0,9
%A A369324 _Stefano Spezia_, Jan 20 2024