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.

1, 7, 1, 71, 23, 1, 913, 456, 48, 1, 13777, 9060, 1560, 82, 1, 233119, 185805, 44262, 3950, 125, 1, 4298911, 3951927, 1188747, 151585, 8355, 177, 1, 84769393, 87024056, 31242008, 5172370, 416730, 15666, 238, 1, 1763748273, 1977448272, 815985408, 165150744, 17626140, 985068, 26936, 308, 1

Offset: 0

Author

Michel Marcus, Oct 31 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
       1;
       7,      1;
      71,     23,     1;
     913,    456,    48,    1;
   13777,   9060,  1560,   82,   1;
  233119, 185805, 44262, 3950, 125, 1;
  ...
		

Links

• Fangfang Cai, Qing-Hu Hou, Yidong Sun, Arthur L.B. Yang, Combinatorial identities related to 2×2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018. See Table 1.3 p. 3.

Crossrefs

Cf. A110440.

Programs

Mathematica

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[n_, k_] := Det[{{s[n, k], s[n, k+1]}, {s[n+1, k], s[n+1, k+1]}}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 02 2019, translated from PARI *)

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
T(n,k) = matdet([s(n,k), s(n,k+1); s(n+1,k), s(n+1,k+1)]);