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 A002190 Sum_{n>=0} a(n)*x^n/n!^2 = -log(BesselJ(0,2*sqrt(x))). (Formerly M3651 N1484) 12
 0, 1, 1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, 229195817258100, 24730000147369440, 3113066087894608560, 452168671458789789504, 75059305956331837485345, 14121026957032156557396000, 2988687741694684876495689040 (list; graph; refs; listen; history; text; internal format)
 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 J.-C. Aval, A. Boussicault, M. Bouvel and M. Silimbani, Combinatorics of non-ambiguous trees, 2012. - From N. J. A. Sloane, Jan 03 2013 Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, Matteo, 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, Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], (2017). L. Carlitz, A sequence of integers related to the Bessel functions, Proc. Amer. Math. Soc., 14 (1963), 1-9. William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], 2017. 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, Kai-Uwe Schmidt, Lq norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016. 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 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 Cf. A101981. A diagonal of A217940. Cf. A115368. Sequence in context: A193421 A179421 * A101981 A002018 A219504 A258180 Adjacent sequences:  A002187 A002188 A002189 * A002191 A002192 A002193 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

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