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%I #5 Dec 27 2021 11:05:34
%S 1,1,0,1,1,0,1,2,0,0,1,3,2,0,0,1,6,6,10,0,0,1,7,16,52,0,0,0,1,8,36,
%T 148,116,8,0,0,1,9,58,448,644,528,12,0,0,1,12,82,885,2932,4032,1326,0,
%U 0,0
%N Array read by antidiagonals: T(n,k) is the number of sequences of length 2*n+1 with terms in 0..k such that the Hankel matrix of the sequence is singular, but the Hankel matrix of any proper subsequence with an odd number of consecutive terms is invertible, n, k >= 0.
%C T(n,2) = 0 for n = 4 and for n >= 7.
%e Array begins:
%e n\k| 0 1 2 3 4 5
%e ---+----------------------
%e 0 | 1 1 1 1 1 1
%e 1 | 0 1 2 3 6 7
%e 2 | 0 0 2 6 16 36
%e 3 | 0 0 10 52 148 448
%e 4 | 0 0 0 116 644 2932
%e For n = 2 and k = 4, the following T(2,4) = 16 sequences are counted:
%e (1, 1, 2, 2, 4),
%e (1, 2, 1, 2, 1),
%e (1, 2, 2, 4, 4),
%e (1, 3, 1, 3, 1),
%e (1, 4, 1, 4, 1),
%e (2, 1, 2, 1, 2),
%e (2, 3, 2, 3, 2),
%e (2, 4, 2, 4, 2),
%e (3, 1, 3, 1, 3),
%e (3, 2, 3, 2, 3),
%e (3, 4, 3, 4, 3),
%e (4, 1, 4, 1, 4),
%e (4, 2, 2, 1, 1),
%e (4, 2, 4, 2, 4),
%e (4, 3, 4, 3, 4),
%e (4, 4, 2, 2, 1).
%Y Cf. A350330, A350364.
%Y Cf. A000012 (row n = 0), A132188 (row n = 1), A000007 (column k = 0), A019590 (column k = 1).
%K nonn,tabl,more
%O 0,8
%A _Pontus von Brömssen_, Dec 27 2021
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