A350571 - OEIS

%I #12 Jan 07 2022 11:15:11

%S 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,

%T 343,338,158,60,20,7,2,1,951,929,457,170,61,20,7,2,1,2615,2598,1286,

%U 498,173,61,20,7,2,1,7318,7261,3678,1421,510,174,61,20,7,2,1

%N 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.

%C It appears that the columns converge to A116950.

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

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

%e Triangle T(n,k) begins:

%e     1;

%e     1,   1;

%e     3,   2,   1;

%e     7,   6,   2,   1;

%e    19,  16,   7,   2,  1;

%e    47,  45,  19,   7,  2,  1;

%e   130, 121,  57,  20,  7,  2, 1;

%e   343, 338, 158,  60, 20,  7, 2, 1;

%e   951, 929, 457, 170, 61, 20, 7, 2, 1;

%e   ...

%t nn = 10; A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt",

%t     "Table"], {_, _}][[;; nn, 2]];

%t A000081 = Drop[Cases[ Import["https://oeis.org/A000081/b000081.txt",

%t      "Table"], {_, _}][[;; nn + 1, 2]], 1];

%t 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

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

%Y Cf. A116950.

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Jan 06 2022