A261002 - OEIS

%I #31 Aug 01 2023 08:05:27

%S 1,2,4,14,66,392,2806,23490,225104,2429538,29159162,385196964,

%T 5553689194,86776290538,1460602599748,26346881876166,507037223792778,

%U 10369313541925896,224565589119822926,5134144053785703610,123569877999804177120,3123085821618415219082

%N Number of unordered unilabeled-bilabeled increasing trees.

%H Vaclav Kotesovec, <a href="/A261002/b261002.txt">Table of n, a(n) for n = 1..435</a> (terms 1..100 from Lars Blomberg)

%H Markus Kuba and Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], 2014.

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

%F 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

%F a(n) = A261001(n) + A261001(n-1). - _Michael Somos_, Jan 17 2017

%F E.g.f. A(x) satisfies A''' = A'' * (A' + A'' / (1 + A')). - _Michael Somos_, Jan 18 2017

%F a(n) ~ d^n * (n-1)!, where d = 1.20456083370247231... - _Vaclav Kotesovec_, Aug 17 2018

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

%t terms = 22;

%t A[_] = 0;

%t Do[A[x_] = Integrate[1 + Integrate[A'[x] (A'[x] + A[x]), x], x] + O[x]^(terms+2), {terms+2}];

%t Rest[CoefficientList[Log[A'[x]], x] * Range[0, terms]!] (* _Jean-François Alcover_, Aug 17 2018, after _Michael Somos_ *)

%t 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 *)

%o (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 */

%Y Cf. A261001.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Aug 09 2015

%E More terms from _Lars Blomberg_, Aug 20 2015