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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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12
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1, 1, 3, 19, 211, 3651, 90921, 3081513, 136407699, 7642177651, 528579161353, 44237263696473, 4405990782649369, 515018848029036937, 69818743428262376523, 10865441556038181291819, 1923889742567310611949459, 384565973956329859109177427
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OFFSET
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0,3
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COMMENTS
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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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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).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..100
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.
Carlitz, L., Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bulletin of the American Mathematical Society 80.5 (1974): 881-884. [Annotated scanned copy]
L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975
Gunnar Thor Magnússon, The inner product on exterior powers of a complex vector space, arXiv preprint arXiv:1401.4048, 2014
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.
Jonathan D. H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67 (1982), no. 1, 173-187.
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.
Sum_{n>=0} a(n) * x^n / n!^2 = 1 / J_0(sqrt(4*x)). - _Michael Somos, May 17 2004
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)
E.g.f.: 1/J_0(sqrt(4*x))= 1 + x/Q(0), where Q(k) = (k+1)^2 - x + (k+1)^2*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2013
a(n) ~ c * (n!)^2 / r^n, where r = 1/4*BesselJZero[0,1]^2 = 1.4457964907366961302939989396139517587678604516 and c = 1/(sqrt(r) * BesselJ(1, 2*sqrt(r))) = 1.60197469692804662664846689139151227422675123376219... - Vaclav Kotesovec, Mar 02 2014, updated Apr 01 2018
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EXAMPLE
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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
G.f. = 1 + x + 3*x^2 + 19*x^3 + 211*x^4 + 3651*x^5 + 90921*x^6 + ...
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MAPLE
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A000275 := proc(n) sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): n!^2*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:
seq(A000275(n), n=0..17); # Peter Luschny, May 27 2017
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MATHEMATICA
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a[0] = 1; a[n_] := a[n] = Sum[(-1)^(r+n+1)*Binomial[n, r]^2 a[r], {r, 0, n-1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 05 2013 *)
CoefficientList[Series[1/BesselJ[0, Sqrt[4*x]], {x, 0, 20}], x]* Range[0, 20]!^2 (* Vaclav Kotesovec, Mar 02 2014 *)
a[ n_] := If[ n < 0, 0, (n! 2^n)^2 SeriesCoefficient[ 1 / BesselJ[ 0, x], {x, 0, 2 n}]]; (* Michael Somos, Aug 20 2015 *)
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PROG
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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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Row 2 of A212855.
Cf. A055133 (absolute value of column 0 of triangle), A192721 (column 1), A115368.
Sequence in context: A049056 A204262 A165356 * A058165 A074707 A230317
Adjacent sequences: A000272 A000273 A000274 * A000276 A000277 A000278
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Christian G. Bower, Apr 25 2000
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STATUS
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approved
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