A127136 Triangle read by rows: T(n,k) is the number of endofunctions on n objects with k components.

1, 2, 1, 4, 2, 1, 9, 7, 2, 1, 20, 17, 7, 2, 1, 51, 48, 21, 7, 2, 1, 125, 127, 60, 21, 7, 2, 1, 329, 352, 174, 65, 21, 7, 2, 1, 862, 963, 504, 190, 65, 21, 7, 2, 1, 2311, 2689, 1456, 570, 196, 65, 21, 7, 2, 1, 6217, 7496, 4212, 1684, 590, 196, 65, 21, 7, 2, 1

Offset

1,2

Comments

For k > n/2, T(n,k) = T(n-1,k-1). - Geoffrey Critzer, Oct 13 2012

Links

Table of n, a(n) for n=1..66.

Formula

G.f.: Product_{k>=1}, 1/(1 - x^k*y)^A002861(k)).

Example

For n = 3, the 7 endofunctions are (1,2,3) -> (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,2,3), (1,3,2) and (2,3,1). The components are respectively 123, 123, 13|2, 123, 1|2|3, 1|23 and 123; the number of components is thus 1, 1, 2, 1, 2, 3, 2, 1, so row 3 is 4,2,1.
The triangle starts:
   1;
   2,  1;
   4,  2,  1;
   9,  7,  2,  1;
  20, 17,  7,  2,  1;

Mathematica

Needs["Combinatorica`"];
nn=30; s[n_, k_]:=s[n, k]=a[n+1-k]+If[n<2 k, 0, s[n-k, k]]; a[1]=1; a[n_]:=a[n]=Sum[a[i] s[n-1, i] i, {i, 1, n-1}]/(n-1); rt=Table[a[i], {i, 1, nn}]; c=Drop[Apply[Plus, Table[Take[CoefficientList[CycleIndex[CyclicGroup[n], s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], nn], {n, 1, 30}]], 1]; CoefficientList[Series[Product[1/(1-y x^i)^c[[i]], {i, 1, nn-1}], {x, 0, 10}], {x, y}]//Grid  (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000081 *)

Crossrefs

Cf. A001372 (row sums), A127124, A127125, A002861 (first column).

Sequence in context: A354802 A158982 A127124 * A239101 A362266 A145983

Adjacent sequences: A127133 A127134 A127135 * A127137 A127138 A127139

Keyword

nonn,tabl

Author

Franklin T. Adams-Watters, Jan 05 2007

Extensions

More terms from Geoffrey Critzer, Oct 13 2012

Corrected and extended by Alois P. Heinz, May 24 2013

Status

approved