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.

1, 1, 1, 6, 2, 1, 42, 18, 3, 1, 416, 168, 36, 4, 1, 5210, 2080, 420, 60, 5, 1, 79212, 31260, 6240, 840, 90, 6, 1, 1417094, 554484, 109410, 14560, 1470, 126, 7, 1, 29168624, 11336752, 2217936, 291760, 29120, 2352, 168, 8, 1, 679100562, 262517616, 51015384, 6653808, 656460, 52416, 3528, 216, 9, 1

Offset

0,4

References

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

Links

Table of n, a(n) for n=0..54.

Formula

E.g.f.: 1/(1-T(x))*exp(T(x)-x+y*x) where T(x) is the e.g.f. for A000169.

Example

      1;
      1,     1;
      6,     2,    1;
     42,    18,    3,   1;
    416,   168,   36,   4,  1;
   5210,  2080,  420,  60,  5, 1;
  79212, 31260, 6240, 840, 90, 6, 1;

Mathematica

nn = 10; t[x_] := Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Map[Select[#, # > 0 &] &,
  Range[0, nn]! CoefficientList[Series[1/(1 - t[ x])*Exp[t[ x] - x + y x], {x, 0, nn}], {x, y}]] // Grid

Crossrefs

Cf. A377535, A000169.

Sequence in context: A101818 A138186 A110321 * A111553 A141473 A068931

Adjacent sequences: A377760 A377761 A377762 * A377764 A377765 A377766

Keyword

nonn,tabl

Author

Geoffrey Critzer, Nov 06 2024

Status

approved