A391206 E.g.f. A(x) satisfies A(x) = 1/(1 - A(x) * Sum_{i>0} x^i/(i! + A(x)*x^i)).

1, 1, 3, 19, 167, 1881, 26655, 455463, 9076919, 206988913, 5322034793, 152333571303, 4804267653047, 165552509055571, 6189393910521307, 249534333735302649, 10792240521968254999, 498441903977024442881, 24484990389453763815201, 1274755126330942972043679

Offset

0,3

Comments

a(n) is the number of ordered rooted trees such that the non-root nodes are each labeled with a subset of [n], no two adjacent sibling nodes are labeled with subsets of the same size, and the labels of a given tree form a set partition of [n].

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f. satisfies A(x) = 1/(1 + Sum_{k>=1} Sum_{m>=1} ( -A(x)*x^k/k! )^m ). - Paul D. Hanna, Mar 14 2026

Example

The tree below has node labels that partition [7] and no adjacent sibling nodes have the same weight so this is counted under a(7) = 455463.
        ___o___
       /   |   \
     {6} {2,3} {5}
         /   \
      {4,7}  {1}
a(2) = 3 counts:
    o        o         o
    |        |         |
  {1,2}     {1}       {2}
             |         |
            {2}       {1}
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 167*x^4/4! + 1881*x^5/5! + 26655*x^6/6! + 455463*x^7/7! + 9076919*x^8/8! + ...

Prog

(PARI) {a(n) = my(A=1, x='x+x*O('x^n)); for(i=1, n,
A = 1/(1 - sum(k=1, n, (A*x^k/k!) / (1 + (A*x^k/k!)) ) )); EGF=A; n!*polcoef(A, n)}
{upto(n) = a(n); Vec(serlaplace(EGF))}
upto(30) \\ Paul D. Hanna, Mar 14 2026

Crossrefs

Cf. A000108, A114902.

Sequence in context: A201123 A260700 A105624 * A238640 A349253 A080835

Adjacent sequences: A391203 A391204 A391205 * A391207 A391208 A391209

Keyword

nonn

Author

John Tyler Rascoe, Mar 08 2026

Status

approved