A360984 Triangular array read by rows. T(n,k) is the number of idempotent Boolean relation matrices on [n] with exactly k reflexive points, n >= 0, 0 <= k <= n.

1, 1, 1, 1, 6, 4, 1, 27, 66, 29, 1, 108, 780, 1116, 355, 1, 405, 8020, 29250, 28405, 6942, 1, 1458, 76110, 649260, 1460425, 1068576, 209527

Offset: 0

Author

Geoffrey Critzer, Feb 27 2023

Keywords

• nonn
• hard
• tabl
• more

Examples

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   6,    4;
  1,  27,   66,    29;
  1, 108,  780,  1116,   355;
  1, 405, 8020, 29250, 28405, 6942;
  ...
		

Crossrefs

Cf. A121337 (row sums), A000798 (main diagonal).

Cf. A245767, A027471 (column 1).

Formula

T(n,n) = A245767(n,n) = A000798(n).

T(n,n-1) = A245767(n,n-1).

T(n,1) = n*Sum_k Sum_j binomial(n-1,k)*binomial(n-1-k,j) = A027471(n+1).

E.g.f. for column 1 is x*exp(x)^3.

E.g.f. for column 2 is x^2/2*exp(x)^3 + x^2*exp(x)^6 + x^2/2*exp(x)^7.

E.g.f. for column 3 is x^3/3!*exp(x)^15 + x^3/3!*exp(x)^3 + x^3*exp(x)^10 + x^3*exp(x)^12 + x^3/2!*exp(x)^7 + 2*x^3/2!*exp(x)^6 + 2*x^3/2*exp(x)^12.

Extensions

Rows 5 and 6 added by Geoffrey Critzer, Mar 05 2023