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Search: id:A000275
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| A000275 |
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Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.
(Formerly M3065 N1242)
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+0
3
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| 1, 1, 3, 19, 211, 3651, 90921, 3081513, 136407699, 7642177651, 528579161353, 44237263696473, 4405990782649369, 515018848029036937, 69818743428262376523, 10865441556038181291819, 1923889742567310611949459, 384565973956329859109177427
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Sum_{n>=0} a(n)*x^n/n!^2 = 1/J_0(sqrt(4x)).
a(n) has the Lucas property, namely a(n) is congruent to a(n_0)a(n_1)...a(n_k) modulo p for any prime p where n_0,n_1,... are the base p digits of n. (Carlitz via McIntosh)
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REFERENCES
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L. Carlitz, The coefficients of the reciprocal of J_0(x), Archiv. Math., 6 (1955), 121ff.
L. Carlitz, R. Scoville and T. Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc., 80 (1974), 881-884.
J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498.
Smith, Jonathan D. H.; Commutative Moufang loops and Bessel functions. Invent. Math. 67 (1982), no. 1, 173-187.
R. McIntosh, A generalization of a congruential property of Lucas, Amer. Math. Monthly 99 (1992), no. 3, 231-238. see page 232. MR1216210 (95b:11008)
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LINKS
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Index entries for sequences related to Bessel functions or polynomials
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FORMULA
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a(n) = Sum (-1)^(r+n+1) binomial(n, r)^2 a(r), r=0..n-1, if n>0.
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PROGRAM
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(PARI) a(n)=if(n<0, 0, n!^2*4^n*polcoeff(1/besselj(0, x+x*O(x^(2*n))), 2*n)) /* Michael Somos May 17 2004 */
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CROSSREFS
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Absolute value of Column 0 of triangle A055133.
Sequence in context: A052886 A079144 A049056 this_sequence A058165 A074707 A135749
Adjacent sequences: A000272 A000273 A000274 this_sequence A000276 A000277 A000278
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KEYWORD
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nonn,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms from Christian G. Bower (bowerc(AT)usa.net), Apr 25 2000
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