0,3

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)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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)

Index entries for sequences related to Bessel functions or polynomials

a(n) = Sum (-1)^(r+n+1) binomial(n, r)^2 a(r), r=0..n-1, if n>0.

(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 */

Absolute value of Column 0 of triangle A055133.

Sequence in context: A079144 A049056 A165356 this_sequence A058165 A074707 A135749

Adjacent sequences: A000272 A000273 A000274 this_sequence A000276 A000277 A000278

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Christian G. Bower (bowerc(AT)usa.net), Apr 25 2000