A261001 - OEIS

%I #31 Jul 31 2023 18:53:09

%S 1,1,3,11,55,337,2469,21021,204083,2225455,26933707,358263257,

%T 5195425937,81580864601,1379021735147,24967860141019,482069363651759,

%U 9887244178274137,214678344941548789,4919465708844154821,118650412290960022299,3004435409327455196783

%N Numbers Q_n arising in computation of number of unordered unilabeled-bilabeled increasing trees.

%H Vaclav Kotesovec, <a href="/A261001/b261001.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'' = A' * (A' + A). - _Michael Somos_, Jan 17 2017

%F a(n) + a(n-1) = A261002(n). - _Michael Somos_, Jan 17 2017

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

%e G.f. = x + x^2 + 3*x^3 + 11*x^4 + 55*x^5 + 337*x^6 + 2469*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+1) // Normal, {terms+1}];

%t Rest[CoefficientList[A[x], x]] Range[terms]! (* _Jean-François Alcover_, Aug 16 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], {n, 1, nmax}] (* _Vaclav Kotesovec_, Aug 17 2018 *)

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

%Y Cf. A261002.

%K nonn

%O 1,3

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

%E More terms from _Lars Blomberg_, Aug 20 2015