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%I M3651 N1484 #83 Jul 15 2021 15:05:41
%S 0,1,1,4,33,456,9460,274800,10643745,530052880,32995478376,
%T 2510382661920,229195817258100,24730000147369440,3113066087894608560,
%U 452168671458789789504,75059305956331837485345,14121026957032156557396000,2988687741694684876495689040
%N Sum_{n>=0} a(n)*x^n/n!^2 = -log(BesselJ(0,2*sqrt(x))).
%C Number of non-ambiguous trees, see the Aval et al. reference. - _Joerg Arndt_, May 11 2015
%D Stany De Smedt, On Sloane's Sequence 1484, Saitama Math. J. 15 (1997), 9-13.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002190/b002190.txt">Table of n, a(n) for n = 0..100</a>
%H J.-C. Aval, A. Boussicault, M. Bouvel and M. Silimbani, <a href="http://www.labri.fr/perso/boussica/archives/publications/combinatorics_of_non_ambiguous_trees.pdf">Combinatorics of non-ambiguous trees</a>, 2012. - From _N. J. A. Sloane_, Jan 03 2013
%H Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, Matteo, <a href="http://arxiv.org/abs/1305.3716">Combinatorics of non-ambiguous trees</a>, arXiv:1305.3716 [math.CO], (16-May-2013).
%H Juan Arias de Reyna, Richard P. Brent and Jan van de Lune, <a href="http://arxiv.org/abs/1205.4423">On the sign of the real part of the Riemann zeta-function</a>, arXiv preprint arXiv:1205.4423 [math.NT], 2012.
%H Beáta Bényi, Gábor V. Nagy, <a href="https://arxiv.org/abs/1707.06899">Bijective enumerations of Γ-free 0-1 matrices</a>, arXiv:1707.06899 [math.CO], (2017).
%H L. Carlitz, <a href="http://dx.doi.org/10.1090/S0002-9939-1963-0166147-X">A sequence of integers related to the Bessel functions</a>, Proc. Amer. Math. Soc., 14 (1963), 1-9.
%H William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, <a href="https://arxiv.org/abs/1702.02446">Tiered trees, weights, and q-Eulerian numbers</a>, arXiv:1702.02446 [math.CO], 2017.
%H Mark Dukes, Thomas Selig, Jason P. Smith, Einar Steingrimsson, <a href="https://arxiv.org/abs/1810.02437">Permutation graphs and the Abelian sandpile model, tiered trees and non-ambiguous binary trees</a>, arXiv:1810.02437 [math.CO], 2018.
%H Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, <a href="http://arxiv.org/abs/math.CO/0606370">A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics</a>, arXiv:math/0606370 [math.CO], 2006.
%H Christian Günther, Kai-Uwe Schmidt, <a href="http://arxiv.org/abs/1602.01750">Lq norms of Fekete and related polynomials</a>, arXiv:1602.01750 [math.NT], 2016.
%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F 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
%F a(n) ~ n! * (n-1)! / r^n, where r = 1/4*BesselJZero[0,1]^2 = 1.44579649073669613... - _Vaclav Kotesovec_, Mar 02 2014
%F 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
%e -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
%p a:= n-> coeff(series(-ln(BesselJ(0,2*sqrt(x))), x, n+1), x, n)*(n!)^2:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 10 2010
%t 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 *)
%t 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 *)
%o (PARI)
%o N=66; x='x+O('x^N);
%o f=-log(sum(n=0,N, (-x)^n/(n!)^2) );
%o f=serlaplace(f);
%o f=serlaplace(f);
%o concat([0],Vec(f))
%o \\ _Joerg Arndt_, May 17 2013
%o (PARI) \\ Terms starting from a(1)=1:
%o N=33; B=vector(N); B[1]=1; b(j)=B[j+1];
%o 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
%o \\ _Joerg Arndt_, May 11 2015
%Y Cf. A101981. A diagonal of A217940.
%Y Cf. A115368.
%K nonn,nice
%O 0,4
%A _N. J. A. Sloane_
%E More terms and better definition from _Vladeta Jovovic_, Jul 16 2006
%E Edited by Assoc. Editors of the OEIS, Oct 12 2010
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