A384105 Triangle read by rows: T(n,k) is the number of binary relations on a set of n objects, exactly k of which are self referencing, 0 <= k <= n.

1, 1, 1, 3, 4, 3, 16, 36, 36, 16, 218, 752, 1104, 752, 218, 9608, 45960, 90416, 90416, 45960, 9608, 1540944, 9133760, 22692704, 30194176, 22692704, 9133760, 1540944, 882033440, 6154473664, 18425858880, 30679088480, 30679088480, 18425858880, 6154473664, 882033440

Offset

0,4

Comments

Also the number of essentially different simple digraphs on a node set A of size n with a distinguished subset B of size k, where elements are indistinguishable within B and within A \ B.

Links

Table of n, a(n) for n=0..35.

Peter Dolland, Python calculation of T(n,k)

Formula

T(n,k) = T(n,n-k).

T(n,0) = T(n,n) = A000273(n).

T(n,1) = T(n,n-1) = A353996(n+1) = A329874(n,4).

Sum_{k=0..n} T(n,k) = A000595(n).

Example

Triangle starts:
            1
            1,              1
            3,              4,              3
           16,             36,             36,              16
          218,            752,           1104,             752,             218
         9608,          45960,          90416,           90416,           45960, ...
      1540944,        9133760,       22692704,        30194176,        22692704, ...
    882033440,     6154473664,    18425858880,     30679088480,     30679088480, ...
1793359192848, 14334221970688, 50138592081152, 100240050239744, 125284653092864, ...
...

Crossrefs

Cf. A000273 (edge cases), A000595 (row sums), A353996, A328874, A383617.

Sequence in context: A342161 A385014 A287199 * A384845 A332830 A288364

Adjacent sequences: A384102 A384103 A384104 * A384106 A384107 A384108

Keyword

nonn,tabl

Author

Peter Dolland, May 19 2025

Status

approved