A002190 Sum_{n>=0} a(n)*x^n/n!^2 = -log(BesselJ(0,2*sqrt(x))).
0, 1, 1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, 229195817258100, 24730000147369440, 3113066087894608560, 452168671458789789504, 75059305956331837485345, 14121026957032156557396000, 2988687741694684876495689040
Offset: 0
Examples
-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
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.
- Mark Dukes, Thomas Selig, Jason P. Smith, 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, 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.
- Index entries for sequences related to Bessel functions or polynomials
Programs
-
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 *) -
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
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
Extensions
More terms and better definition from Vladeta Jovovic, Jul 16 2006
Edited by Assoc. Editors of the OEIS, Oct 12 2010
Comments