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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.

L. Carlitz, A sequence of integers related to the Bessel functions, Proc. Amer. Math. Soc., 14 (1963), 1-9.

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.

Index entries for sequences related to Bessel functions or polynomials

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

N. J. A. Sloane

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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Last modified March 26 10:42 EDT 2017. Contains 284111 sequences.