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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.

%I A350571 #12 Jan 07 2022 11:15:11
%S A350571 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 A350571 343,338,158,60,20,7,2,1,951,929,457,170,61,20,7,2,1,2615,2598,1286,
%U A350571 498,173,61,20,7,2,1,7318,7261,3678,1421,510,174,61,20,7,2,1
%N 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.
%C A350571 It appears that the columns converge to A116950.
%D A350571 O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009.
%F A350571 G.f.: Product_{i>=1} 1/(1-y*x^i)^A000081(i)*Product_{i>=1} 1/(1-x^i)^A002861(i).
%e A350571 Triangle T(n,k) begins:
%e A350571     1;
%e A350571     1,   1;
%e A350571     3,   2,   1;
%e A350571     7,   6,   2,   1;
%e A350571    19,  16,   7,   2,  1;
%e A350571    47,  45,  19,   7,  2,  1;
%e A350571   130, 121,  57,  20,  7,  2, 1;
%e A350571   343, 338, 158,  60, 20,  7, 2, 1;
%e A350571   951, 929, 457, 170, 61, 20, 7, 2, 1;
%e A350571   ...
%t A350571 nn = 10; A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt",
%t A350571     "Table"], {_, _}][[;; nn, 2]];
%t A350571 A000081 = Drop[Cases[ Import["https://oeis.org/A000081/b000081.txt",
%t A350571      "Table"], {_, _}][[;; nn + 1, 2]], 1];
%t A350571 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 A350571 Cf. A126285 (row sums), A001372 (column k=0), A000081, A002861.
%Y A350571 Cf. A116950.
%K A350571 nonn,tabl
%O A350571 0,4
%A A350571 _Geoffrey Critzer_, Jan 06 2022