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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 09 2015
EXTENSIONS
More terms from Lars Blomberg, Aug 20 2015
STATUS
approved