A201685 Triangular array read by rows. T(n,k) is the number of connected endofunctions on {1,2,...,n} that have exactly k nodes in the unique cycle of its digraph representation.

1, 2, 1, 9, 6, 2, 64, 48, 24, 6, 625, 500, 300, 120, 24, 7776, 6480, 4320, 2160, 720, 120, 117649, 100842, 72030, 41160, 17640, 5040, 720, 2097152, 1835008, 1376256, 860160, 430080, 161280, 40320, 5040, 43046721, 38263752, 29760696, 19840464, 11022480, 4898880, 1632960, 362880, 40320

Offset

1,2

Comments

Column k=1: A000169,

Column k=2: A053506,

Column k=3: A065513.

Row sums: A001865.

T(n,n) = (n-1)!, T(n,n-1) = n!.

Sum_{k=1..n} T(n,k)*k = n^n. - Geoffrey Critzer, May 13 2013

From the asymptotic given by N-E. Fahssi in A001865, we see the expected size of the cycle grows as (2*n/Pi)^(1/2). - Geoffrey Critzer, May 13 2013

Central terms: A277168. - Paul D. Hanna, Oct 01 2016

Links

Alois P. Heinz, Rows n = 1..141, flattened

Formula

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

T(n,k) = binomial(n-1,k-1)*n^(n-k)*(k-1)!. - Geoffrey Critzer, May 13 2013

Example

Triangle begins as:
     1;
     2,    1;
     9,    6,    2;
    64,   48,   24,    6;
   625,  500,  300,  120,  24;
  7776, 6480, 4320, 2160, 720, 120;

Maple

T:= (n, k)-> binomial(n-1, k-1)*n^(n-k)*(k-1)!:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Aug 14 2013

Mathematica

f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Map[f, Drop[Range[0, 10]! CoefficientList[Series[Log[1/(1 - y t)], {x, 0, 10}], {x, y}], 1]] // Grid

Prog

(PARI) T(n, k) = binomial(n-1, k-1)*n^(n-k)*(k-1)!; \\ G. C. Greubel, Jan 08 2020
(Magma) [Binomial(n-1, k-1)*n^(n-k)*Factorial(k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jan 08 2020
(Sage) [[binomial(n-1, k-1)*n^(n-k)*factorial(k-1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jan 08 2020
(GAP) Flat(List([1..12], n-> List([1..n], k-> Binomial(n-1, k-1)*n^(n-k)*Factorial(k-1) ))); # G. C. Greubel, Jan 08 2020

Crossrefs

Cf. A000169, A001865, A053506, A065513, A277168.

Sequence in context: A061356 A141028 A246323 * A120671 A245461 A086572

Adjacent sequences: A201682 A201683 A201684 * A201686 A201687 A201688

Keyword

nonn,tabl

Author

Geoffrey Critzer, Dec 03 2011

Status

approved