A000275 Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden. (Formerly M3065 N1242)

1, 1, 3, 19, 211, 3651, 90921, 3081513, 136407699, 7642177651, 528579161353, 44237263696473, 4405990782649369, 515018848029036937, 69818743428262376523, 10865441556038181291819, 1923889742567310611949459, 384565973956329859109177427, 86180438505835750284241676121

Offset

0,3

Comments

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

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..261

Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) (with t=0 and m=2) on p. 249.

Leonid Bedratyuk and Nataliia Luno, Connection problems for the generalized hypergeometric Appell polynomials, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 10-18.

L. Carlitz, The coefficients of the reciprocal of J_0(x), Archiv. Math. 6 (1955), 121-127.

L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884.

L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884. [Annotated scanned copy]

L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975.

Jan Geuenich, Tilting modules for the Auslander algebra of K[x]/(xn), arXiv:1803.10707 [math.RT], 2018.

Gunnar Thor Magnússon, The inner product on exterior powers of a complex vector space, arXiv preprint arXiv:1401.4048 [math.AG], 2014.

R. McIntosh, A generalization of a congruential property of Lucas, Amer. Math. Monthly 99(3) (1992), 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(1) (1982), 173-187.

Index entries for sequences related to Bessel functions or polynomials

Formula

a(n) = Sum_{r=0..n-1} (-1)^(r+n+1) binomial(n, r)^2 a(r), 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

Example

From Peter Bala, Aug 08 2011: (Start)
a(3) = 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     0     1     0     1     0
  321|  0     0     0     0     0     0
(End)
G.f. = 1 + x + 3*x^2 + 19*x^3 + 211*x^4 + 3651*x^5 + 90921*x^6 + ...

Maple

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

Mathematica

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 *)

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

Row 2 of A212855.

Cf. A055133 (absolute value of column 0 of triangle), A192721 (column 1), A115368.

Column k=1 of A340986.

Sequence in context: A049056 A204262 A165356 * A058165 A074707 A230317

Adjacent sequences: A000272 A000273 A000274 * A000276 A000277 A000278

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Apr 25 2000

Status

approved