A349950 Triangular array read by rows: T(n,k) is the number of partial functions on [n] with exactly k connected components, n>=0, 0<=k<=n.

1, 0, 2, 0, 5, 4, 0, 26, 30, 8, 0, 206, 283, 120, 16, 0, 2194, 3360, 1790, 400, 32, 0, 29352, 48538, 29835, 8660, 1200, 64, 0, 472730, 828758, 563486, 193130, 35560, 3360, 128, 0, 8902448, 16352684, 12000604, 4628057, 1023120, 130592, 8960, 256, 0, 191915874, 366387696, 285672572, 120489264, 30357474, 4711392, 442176, 23040, 512

Offset

0,3

Comments

Row sums equal (n+1)^n, the number of partial functions on [n].

Links

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

Formula

E.g.f.: exp(y*log(f(x))) where f(x) = 1/(1-t(x))*exp(t(x) and t(x) is the e.g.f. for A000169.

Example

  1;
  0,    2;
  0,    5,    4;
  0,   26,   30,    8;
  0,  206,  283,  120,  16;
  0, 2194, 3360, 1790, 400, 32;
  ...

Mathematica

nn = 10; t[x_] := Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; f[x_] := 1/(1 -t[x])Exp[t[x]]; Table[(Range[0, nn]! CoefficientList[ Series[Exp[y Log[f[x]]], {x, 0, nn}], {x, y}])[[i, 1 ;; i]], {i, 1, nn}] // Grid

Crossrefs

Cf. A000169 (row sums).

Sequence in context: A324245 A173732 A086280 * A164976 A261745 A083714

Adjacent sequences: A349947 A349948 A349949 * A349951 A349952 A349953

Keyword

nonn,tabl

Author

Geoffrey Critzer, Dec 06 2021

Status

approved