A350571 Triangular array read by rows. T(n,k) is the number of unlabeled partial functions on [n] with exactly k undefined points, n>=0, 0<=k<=n.

1, 1, 1, 3, 2, 1, 7, 6, 2, 1, 19, 16, 7, 2, 1, 47, 45, 19, 7, 2, 1, 130, 121, 57, 20, 7, 2, 1, 343, 338, 158, 60, 20, 7, 2, 1, 951, 929, 457, 170, 61, 20, 7, 2, 1, 2615, 2598, 1286, 498, 173, 61, 20, 7, 2, 1, 7318, 7261, 3678, 1421, 510, 174, 61, 20, 7, 2, 1

Offset: 0

Author

Geoffrey Critzer, Jan 06 2022

Keywords

• nonn
• tabl

Comments

It appears that the columns converge to A116950.

Examples

			Triangle T(n,k) begins:
    1;
    1,   1;
    3,   2,   1;
    7,   6,   2,   1;
   19,  16,   7,   2,  1;
   47,  45,  19,   7,  2,  1;
  130, 121,  57,  20,  7,  2, 1;
  343, 338, 158,  60, 20,  7, 2, 1;
  951, 929, 457, 170, 61, 20, 7, 2, 1;
  ...
		

References

• O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009.

Crossrefs

Cf. A126285 (row sums), A001372 (column k=0), A000081, A002861.

Cf. A116950.

Programs

Mathematica

nn = 10; A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt",
    "Table"], {, }][[;; nn, 2]];
A000081 = Drop[Cases[ Import["https://oeis.org/A000081/b000081.txt",
     "Table"], {, }][[;; nn + 1, 2]], 1];
Map[Select[#, # > 0 &] &, CoefficientList[Series[ Product[1/(1 -  y x^i)^A000081[[i]], {i, 1, nn}] Product[1/(1 - x^i)^A002861[[i]], {i, 1, nn}], {x, 0, nn}], {x,y}]] // Grid

Formula

G.f.: Product_{i>=1} 1/(1-y*x^i)^A000081(i)*Product_{i>=1} 1/(1-x^i)^A002861(i).