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A321187 Triangle read by rows, T(n, k) is the determinant of the matrix [s(n,k), s(n,k+1); s(n+1,k), s(n+1,k+1)] where s is the triangle A110440 of little Schroeder numbers.

%I A321187 #25 Feb 02 2019 10:30:45
%S A321187 1,7,1,71,23,1,913,456,48,1,13777,9060,1560,82,1,233119,185805,44262,
%T A321187 3950,125,1,4298911,3951927,1188747,151585,8355,177,1,84769393,
%U A321187 87024056,31242008,5172370,416730,15666,238,1,1763748273,1977448272,815985408,165150744,17626140,985068,26936,308,1
%N A321187 Triangle read by rows, T(n, k) is the determinant of the matrix [s(n,k), s(n,k+1); s(n+1,k), s(n+1,k+1)] where s is the triangle A110440 of little Schroeder numbers.
%H A321187 Fangfang Cai, Qing-Hu Hou, Yidong Sun, Arthur L.B. Yang, <a href="https://arxiv.org/abs/1808.05736">Combinatorial identities related to 2×2 submatrices of recursive matrices</a>, arXiv:1808.05736 [math.CO], 2018. See Table 1.3 p. 3.
%e A321187 Triangle begins:
%e A321187        1;
%e A321187        7,      1;
%e A321187       71,     23,     1;
%e A321187      913,    456,    48,    1;
%e A321187    13777,   9060,  1560,   82,   1;
%e A321187   233119, 185805, 44262, 3950, 125, 1;
%e A321187   ...
%t A321187 s[n_, k_] := Sum[i (-1)^(k - i + 1) Binomial[k + 1, i] Sum[(-1)^j 2^(n + 1 - j) (2n + i - j + 1)!/((n + i - j + 1)! j! (n - j + 1)!), {j, 0, n+1}], {i, 0, k + 1}];
%t A321187 T[n_, k_] := Det[{{s[n, k], s[n, k+1]}, {s[n+1, k], s[n+1, k+1]}}];
%t A321187 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 02 2019, translated from PARI *)
%o A321187 (PARI) s(n,k) = sum(i = 0, k+1, (i*(-1)^(k - i + 1)*binomial(k + 1, i)*sum(j=0, n+1, (-1)^j*2^(n + 1 - j)*(2*n + i - j + 1)!/((n + i - j + 1)!*j!*(n - j + 1)!)))); \\ A110440
%o A321187 T(n,k) = matdet([s(n,k), s(n,k+1); s(n+1,k), s(n+1,k+1)]);
%Y A321187 Cf. A110440.
%K A321187 nonn,tabl
%O A321187 0,2
%A A321187 _Michel Marcus_, Oct 31 2018