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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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)

Examples

			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 + ...
		

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

Crossrefs

Row 2 of A212855.

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

Column k=1 of A340986.

Programs

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 *)

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 */

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

Extensions

More terms from Christian G. Bower, Apr 25 2000