A206823 -id:A206823 - cp's OEIS frontend

A231797 Number of endofunctions on [n] such that no element has a preimage of cardinality one.

1, 0, 2, 3, 40, 205, 2556, 24409, 347712, 4794633, 81163900, 1434596581, 28725779952, 612610306477, 14280306222924, 354958921699425, 9471543095892736, 268347925179992593, 8075532017006497404, 256672899448317943453, 8603440900030816095600

Offset: 0

Alois P. Heinz, Nov 13 2013

nonn

			a(2) = 2: (1,1), (2,2).
a(3) = 3: (1,1,1), (2,2,2), (3,3,3).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..410 (first 181 terms from Alois P. Heinz)
R. K. Guy, Letter to N. J. A. Sloane, Jun 21, 1975

Column k=0 of A206823.
A diagonal of A241580. Cf. also A241581.
Column k=1 of A245405.
Cf. A245496.

with(combinat):
b:= proc(t, i, u) option remember; `if`(t=0, 1,
      `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
      *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);

multinomial[n_, k_List] := n!/Times @@ (k!); b[t_, i_, u_] := b[t, i, u] = If[t == 0, 1, If[i < 2, 0, b[t, i - 1, u] + Sum[multinomial[t, Join[{ t - i*j}, Array[i &, j]]] * b[t - i*j, i - 1, u - j]*u!/(u - j)!/j!, {j, 1, t/i}]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)
Table[n!*SeriesCoefficient[(E^x-x)^n,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 23 2014 *)
Flatten[{1,Table[(-1)^n*n! + Sum[Binomial[n,j]^2*(-1)^j*(n-j)^(n-j)*j!,{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 25 2014 *)

{a(n) = n! * polcoeff((exp(x + x*O(x^n)) - x)^n, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Jan 30 2015

a(n) = n! * [x^n] (exp(x)-x)^n.
a(n) ~ (1-exp(-1))^(n+1/2) * n^n. - Vaclav Kotesovec, Jul 23 2014
E.g.f.: 1 / ((1 + x) * (1 + LambertW(-x/(1 + x)))). - Ilya Gutkovskiy, Jan 25 2020
a(n) = Sum_{k=0..n} (-1)^(n-k)*(n-k)!*k^k*binomial(n,k)^2. - Ridouane Oudra, Jul 14 2025

A231602 Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.

Original entry on oeis.org

1, 0, 2, 3, 0, 6, 4, 36, 0, 24, 65, 80, 360, 0, 120, 306, 1950, 1200, 3600, 0, 720, 4207, 12852, 40950, 16800, 37800, 0, 5040, 38424, 235592, 359856, 764400, 235200, 423360, 0, 40320, 573057, 2766528, 8481312, 8636544, 13759200, 3386880, 5080320, 0, 362880

Offset: 1

Geoffrey Critzer, Nov 11 2013

T(n,k) is also the number of functions f:{1,2,...,n-1}->{1,2,...,n} that have exactly k elements whose preimage has cardinality = 1.
T(n,n-1) = n! = A000142(n).
Column k = 0 = A060356(n).
Row sums = n^(n-1) = A000169(n).
Refinement given by A248120. Sum coefficients of the partition polynomials with h_1 = (1') = t and all other h_n = (n') = 1 to obtain this entry. - Tom Copeland, Feb 01 2016

			1;
0, 2;
3, 0, 6;
4, 36, 0, 24;
65, 80, 360, 0, 120;
306, 1950, 1200, 3600, 0, 720;
4207, 12852, 40950, 16800, 37800, 0, 5040;
38424, 235592, 359856, 764400, 235200, 423360, 0, 40320;
....0..........0........
....|........./ \.......
....0........0...0......
.../ \.......|..........
..0   0......0..........
T(4,1) = 36.  Both of these graphs on 4 nodes have exactly 1 node that has outdegree = 1.  There are 12 + 24 = 36 labelings.

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

Cf. A055302, A206823.

Cf. A248120.

with(combinat): C:= binomial:
b:= proc(t, i, u) option remember; `if`(t=0, 1,
      `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
      *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
    end:
T:= (n, k)-> C(n, k)*C(n-1, k)*k! *b(n-1-k$2, n-k):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Nov 12 2013

nn=8;Table[Table[Drop[Range[0,nn]!CoefficientList[Series[-ProductLog[x/(-1-x+x y)],{x,0,nn}],{x,y}],1][[r,c]],{c,1,r}],{r,1,nn}]//Grid

E.g.f. satisfies A(x,y) = y*x*A(x,y) + x*( exp(A(x,y)) - A(x,y) ).

cp's OEIS Frontend

A231797 Number of endofunctions on [n] such that no element has a preimage of cardinality one.

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