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A377763 Triangular array read by rows. T(n,k) is the number of partial functions f on [n] such that there are exactly k points in [n] that are neither in the domain of f nor in the image of f, n>=0, 0<=k<=n.

%I A377763 #9 Nov 07 2024 09:51:51
%S A377763 1,1,1,6,2,1,42,18,3,1,416,168,36,4,1,5210,2080,420,60,5,1,79212,
%T A377763 31260,6240,840,90,6,1,1417094,554484,109410,14560,1470,126,7,1,
%U A377763 29168624,11336752,2217936,291760,29120,2352,168,8,1,679100562,262517616,51015384,6653808,656460,52416,3528,216,9,1
%N A377763 Triangular array read by rows.  T(n,k) is the number of partial functions f on [n] such that there are exactly k points in [n] that are neither in the domain of f nor in the image of f, n>=0, 0<=k<=n.
%D A377763 O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009.
%F A377763 E.g.f.: 1/(1-T(x))*exp(T(x)-x+y*x) where T(x) is the e.g.f. for A000169.
%e A377763       1;
%e A377763       1,     1;
%e A377763       6,     2,    1;
%e A377763      42,    18,    3,   1;
%e A377763     416,   168,   36,   4,  1;
%e A377763    5210,  2080,  420,  60,  5, 1;
%e A377763   79212, 31260, 6240, 840, 90, 6, 1;
%t A377763 nn = 10; t[x_] := Sum[n^(n - 1) x^n/n!, {n, 1, nn}];Map[Select[#, # > 0 &] &,
%t A377763   Range[0, nn]! CoefficientList[Series[1/(1 - t[ x])*Exp[t[ x] - x + y x], {x, 0, nn}], {x, y}]] // Grid
%Y A377763 Cf. A377535, A000169.
%K A377763 nonn,tabl
%O A377763 0,4
%A A377763 _Geoffrey Critzer_, Nov 06 2024