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

0,2

Links

Table of n, a(n) for n=0..44.

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.

Example

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

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 *)

Prog

(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)]);

Crossrefs

Cf. A110440.

Sequence in context: A350202 A237111 A281620 * A221367 A110788 A100254

Adjacent sequences: A321184 A321185 A321186 * A321188 A321189 A321190

Keyword

nonn,tabl

Author

Michel Marcus, Oct 31 2018

Status

approved