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A002190
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Sum_{n>=0} a(n)*x^n/n!^2 = -ln(BesselJ(0,2*sqrt(x))).
(Formerly M3651 N1484)
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7
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0, 1, 1, 4, 33, 456, 9460, 274800, 10643745, 530052880, 32995478376, 2510382661920, 229195817258100, 24730000147369440, 3113066087894608560, 452168671458789789504, 75059305956331837485345, 14121026957032156557396000, 2988687741694684876495689040
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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REFERENCES
| L. Carlitz, A sequence of integers related to the Bessel functions, Proc. Amewr. Math. Soc., 14 (1963), 1-9.
Philippe Flajolet, Eric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math.CO/0606370
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).
Stany De Smedt, On Sloane's Sequence 1484, Saitama Math. J. 15 (1997), 9-13.
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LINKS
| Index entries for sequences related to Bessel functions or polynomials
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FORMULA
| Conjecture: G.f.: 1 = Sum_{n>=0} a(n+1)*A000108(n)*x^n*Sum_{k>=0} C(2n+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(2n+k,k)*(-x)^k. [Paul Hanna, Oct 10 2010]
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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 Hanna, Oct 09 2010]
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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 Heinz, Oct 10 2010]
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MATHEMATICA
| nn=18; CoefficientList[Series[-Log[BesselJ[0, 2*Sqrt[x]]], {x, 0, nn}], x]*Table[n!^2, {n, 0, nn}] (* From Jean-François Alcover, Jun 22 2011 *)
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CROSSREFS
| Cf. A101981.
Sequence in context: A193421 A179421 * A101981 A002018 A072754 A113086
Adjacent sequences: A002187 A002188 A002189 * A002191 A002192 A002193
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms and better definition from Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2006
Edited by Assoc. Editors of OEIS, Oct 12 2010
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