OFFSET
0,4
COMMENTS
Number of non-ambiguous trees, see the Aval et al. reference. - Joerg Arndt, May 11 2015
REFERENCES
Stany De Smedt, On Sloane's Sequence 1484, Saitama Math. J. 15 (1997), 9-13.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..100
Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, and Matteo Silimbani, Combinatorics of non-ambiguous trees, 2012. - From N. J. A. Sloane, Jan 03 2013
Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, and Matteo Silimbani, Combinatorics of non-ambiguous trees, arXiv:1305.3716 [math.CO], (16-May-2013).
Juan Arias de Reyna, Richard P. Brent and Jan van de Lune, On the sign of the real part of the Riemann zeta-function, arXiv preprint arXiv:1205.4423 [math.NT], 2012.
Beáta Bényi and Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], (2017).
Leonard Carlitz, A sequence of integers related to the Bessel functions, Proc. Amer. Math. Soc., 14 (1963), 1-9.
Theo Douvropoulos and Matthieu Josuat-Vergès, Homology character of the parabolic coset poset, arXiv:2509.11905 [math.CO], 2025. See p. 17.
William Dugan, Sam Glennon, Paul E. Gunnells, and Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], 2017.
Mark Dukes, Thomas Selig, Jason P. Smith, and Einar Steingrimsson, Permutation graphs and the Abelian sandpile model, tiered trees and non-ambiguous binary trees, arXiv:1810.02437 [math.CO], 2018.
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario, and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO], 2006.
Christian Günther and Kai-Uwe Schmidt, Lq norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016.
Joshua F. Robinson and Patrick B. Warren, Renormalisation group theory applied to x^.. + x^. + x^2 = 0, arXiv:2412.16578 [math-ph], 2024. See p. 5.
FORMULA
Conjecture: G.f.: 1 = Sum_{n>=0} a(n+1)*A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)^2*(-x)^k. Compare with the following g.f of the Catalan numbers (A000108): 1 = Sum_{n>=0} A000108(n)*x^n*Sum_{k>=0} C(2*n+k,k)*(-x)^k. - Paul D. Hanna, Oct 10 2010
a(n) ~ n! * (n-1)! / r^n, where r = 1/4*BesselJZero[0,1]^2 = 1.44579649073669613... - Vaclav Kotesovec, Mar 02 2014
a(0) = 0; a(n) = -(-1)^n + (1/n) * Sum_{k=1..n-1} (-1)^(n-k-1) * binomial(n,k)^2 * k * a(k). - Ilya Gutkovskiy, Jul 15 2021
EXAMPLE
-log( Sum_{n>=0} (-x)^n/n!^2 ) = x + x^2/2!^2 + 4*x^3/3!^2 + 33*x^4/4!^2 + 456*x^5/5!^2 + 9460*x^6/6!^2 + ... . -Paul D. Hanna, Oct 09 2010
MAPLE
a:= n-> coeff(series(-ln(BesselJ(0, 2*sqrt(x))), x, n+1), x, n)*(n!)^2:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 10 2010
MATHEMATICA
nn=18; CoefficientList[Series[-Log[BesselJ[0, 2*Sqrt[x]]], {x, 0, nn}], x]*Table[n!^2, {n, 0, nn}] (* Jean-François Alcover, Jun 22 2011 *)
Clear[q]; q[n_, 1] := (n-1)!^2; q[n_, k_] := q[n, k] = Sum[Binomial[n-1, j]*Binomial[n-1, j+1]*Sum[q[j+1, r]*q[n-j-1, k-r], {r, Max[1, -n+j+k+1], Min[j+1, k-1]}], { n-2}]; a[n_] := q[n, n]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 13 2013 *)
PROG
(PARI)
N=66; x='x+O('x^N);
f=-log(sum(n=0, N, (-x)^n/(n!)^2) );
f=serlaplace(f);
f=serlaplace(f);
concat([0], Vec(f))
\\ Joerg Arndt, May 17 2013
(PARI) \\ Terms starting from a(1)=1:
N=33; B=vector(N); B[1]=1; b(j)=B[j+1];
for(n=0, N-2, B[n+2]=sum(i=0, n, my(j=n-i); binomial(n+1, i)*binomial(n+1, j)*b(i)*b(j) ) ); B
\\ Joerg Arndt, May 11 2015
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms and better definition from Vladeta Jovovic, Jul 16 2006
Edited by Assoc. Editors of the OEIS, Oct 12 2010
STATUS
approved