A261002 Number of unordered unilabeled-bilabeled increasing trees.

1, 2, 4, 14, 66, 392, 2806, 23490, 225104, 2429538, 29159162, 385196964, 5553689194, 86776290538, 1460602599748, 26346881876166, 507037223792778, 10369313541925896, 224565589119822926, 5134144053785703610, 123569877999804177120, 3123085821618415219082

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..435 (terms 1..100 from Lars Blomberg)

Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.

Formula

Kuba et al. (2014) give a recurrence (see Th. 12).

E.g.f. A(x) satisfies A(x) = log(B'(x)) where B() is the e.g.f. of A261001. - Michael Somos, Jan 17 2017

a(n) = A261001(n) + A261001(n-1). - Michael Somos, Jan 17 2017

E.g.f. A(x) satisfies A''' = A'' * (A' + A'' / (1 + A')). - Michael Somos, Jan 18 2017

a(n) ~ d^n * (n-1)!, where d = 1.20456083370247231... - Vaclav Kotesovec, Aug 17 2018

Example

G.f. = x + 2*x^2 + 4*x^3 + 14*x^4 + 66*x^5 + 392*x^6 + 2806*x^7 + ...

Mathematica

terms = 22;
A[_] = 0;
Do[A[x_] = Integrate[1 + Integrate[A'[x] (A'[x] + A[x]), x], x] + O[x]^(terms+2), {terms+2}];
Rest[CoefficientList[Log[A'[x]], x] * Range[0, terms]!] (* Jean-François Alcover, Aug 17 2018, after Michael Somos *)
nmax = 25; Q[0] = 0; Q[1] = 1; Do[Q[m] = Sum[Binomial[m - 2, k]*(Q[k] + Q[k + 1])*Q[m - k - 1], {k, 0, m - 2}], {m, 2, nmax}]; Table[Q[n] + Q[n-1], {n, 1, nmax}] (* Vaclav Kotesovec, Aug 17 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=0, n, A = intformal( 1 + intformal( A' * (A' + A)))); n! * polcoeff( log(A'), n))}; /* Michael Somos, Jan 18 2017 */

Crossrefs

Cf. A261001.

Sequence in context: A322623 A245139 A020131 * A032147 A007712 A192815

Adjacent sequences: A260999 A261000 A261001 * A261003 A261004 A261005

Keyword

nonn

Author

N. J. A. Sloane, Aug 09 2015

Extensions

More terms from Lars Blomberg, Aug 20 2015

Status

approved