A231602 -id:A231602 - cp's OEIS frontend

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

0, 1, 0, 3, 4, 65, 306, 4207, 38424, 573057, 7753510, 134046671, 2353898196, 47602871329, 1013794852266, 23751106404495, 590663769125296, 15806094859299329, 448284980183376078, 13515502344669830287

Offset: 0

Vladeta Jovovic, Apr 01 2001

Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - Gus Wiseman, Jan 20 2020

			From _Gus Wiseman_, Dec 31 2019: (Start)
Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
  1[2,3[4,5[6,7]]]
  1[2,3[4,5,6,7]]
  1[2[3,4],5[6,7]]
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A052871, A060313.
Cf. A008297.
Column k=0 of A231602.
The unlabeled version is A001678(n + 1).
The case where the root is fixed is A108919.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees with labeled leaves are A000311.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.

List([0..20],n->Sum([1..n],k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1,k-1)*k^(k-1))); # Muniru A Asiru, Feb 19 2018

seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # G. C. Greubel, Mar 16 2020

CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];
Array[a,10] (* Gus Wiseman, Dec 31 2019 *)

{ for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009

my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ G. C. Greubel, Feb 19 2018

a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)*A058863(k). - Vladeta Jovovic, Sep 17 2003
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Nov 27 2012
a(n) = n * A108919(n). - Gus Wiseman, Dec 31 2019

A206823 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 3, 18, 0, 6, 40, 48, 144, 0, 24, 205, 1000, 600, 1200, 0, 120, 2556, 7380, 18000, 7200, 10800, 0, 720, 24409, 125244, 180810, 294000, 88200, 105840, 0, 5040, 347712, 1562176, 4007808, 3857280, 4704000, 1128960, 1128960, 0, 40320

Offset: 0

Geoffrey Critzer, Feb 12 2012

Row sums = n^n, all functions f:{1,2,...,n}->{1,2,...,n}.

T(n,n)= n!, bijections on {1,2,...,n}.

			Triangle T(n,k) begins:
    1;
    0      1;
    2      0     2;
    3     18     0      6;
   40     48   144      0    24;
  205   1000   600   1200     0     120;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

Row sums give: A000312.
Column k=0 gives: A231797.
Cf. A231602.

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, k)*k! *b(n-k$2, n-k):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Nov 13 2013

nn = 8; Prepend[CoefficientList[Table[n! Coefficient[Series[(Exp[x] - x + y x)^n, {x, 0, nn}], x^n], {n, 1, nn}], y], {1}] // Flatten

E.g.f.: Sum_{k=0..n} T(n,k) * y^k * x^n / n! = (exp(x) - x + y*x)^n.

cp's OEIS Frontend

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

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