|
| |
|
|
A000275
|
|
Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.
(Formerly M3065 N1242)
|
|
4
| |
|
|
1, 1, 3, 19, 211, 3651, 90921, 3081513, 136407699, 7642177651, 528579161353, 44237263696473, 4405990782649369, 515018848029036937, 69818743428262376523, 10865441556038181291819, 1923889742567310611949459, 384565973956329859109177427
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| 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)
|
|
|
REFERENCES
| 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.
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)
J. Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498.
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).
Smith, Jonathan D. H.; Commutative Moufang loops and Bessel functions. Invent. Math. 67 (1982), no. 1, 173-187.
|
|
|
LINKS
| Index entries for sequences related to Bessel functions or polynomials
|
|
|
FORMULA
| a(n) = Sum (-1)^(r+n+1) binomial(n, r)^2 a(r), r=0..n-1, if n>0.
From Peter Bala, Aug 08 2011: (Start)
Conjectural formula: 1 = sum {n>=0} a(n)*x^n*sum{k>=0} binomial(n+k,k)^2*(-x)^k.
Apart from the initial term, first column of A192721. (End)
|
|
|
EXAMPLE
| a(2) = 19: The 19 pairs of permutations in the group S_3 x S_3 with
no common rises correspond to the zero entries in the table below.
======================================
.Number of common rises in S_3 x S_3..
======================================
...|.123...132...213...231...312...321
======================================
123|..2.....1.....1.....1.....1.....0
132|..1.....1.....0.....1.....0.....0
213|..1.....0.....1.....0.....1.....0
231|..1.....1.....0.....1.....0.....0
312|..1.....1.....0.....1.....0.....0
321|..0.....0.....0.....0.....0.....0
- Peter Bala Aug 08 2011
|
|
|
PROG
| (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 */
|
|
|
CROSSREFS
| Cf. A055133 (absolute value of column 0 of triangle), A192721 (column 1).
Sequence in context: A049056 A204262 A165356 * A058165 A074707 A135749
Adjacent sequences: A000272 A000273 A000274 * A000276 A000277 A000278
|
|
|
KEYWORD
| nonn,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net), Apr 25 2000
|
| |
|
|
