OFFSET
1,3
COMMENTS
Integer sequence given between equations (16) and (17) of Bender et al., p. 4. A recursion is found for coefficients of Taylor series of r-th powers of generalized Bessel functions.
A001263^(-1) * [1, 2, 3, ...] = A103364 * [1, 2, 3, ...] = (1, 1, -1, 3, -16, 130, -1485, 22645, ...); where A001263 = the Narayana triangle. - Gary W. Adamson, Jan 02 2008
Image of n^2 under A001263^(-1), i.e., A001263^(-1) *[0,1,4,9,...] is [0, 1, 1, -3, 16, -130, 1485, -22645, 444136, ...]. - Paul Barry, Jul 13 2009
LINKS
Carl M. Bender, Dorje C. Brody, and Bernhard K. Meister, On powers of Bessel functions, J. Math. Phys. vol 44, No. 1 (2003) pp 309-314.
Yan Hong, Bai-Ni Guo, and Feng Qi, Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind, Computer Modeling in Engineering and Sciences (2021) Vol. 129, No. 1, 409-423.
F. T. Howard, Integers Related to the Bessel Function J1(z), Fibonacci Quarterly, Volume 23, Number 3, August 1985, pp. 249-257.
FORMULA
For n>1, a(n) = (Sum_{r=1..n-1} binomial(n+1,r+1)*binomial(n+1,r)*a(r)*a(n-r))/(n+1)^2. - Michel Marcus, Oct 17 2012
MAPLE
A131490 := proc(n) local twos, resul; resul := twos*taylor(BesselI(0, twos), twos=0, 2*n+3) ; resul := resul/taylor(BesselI(1, twos), twos=0, 2*n+3) ; resul := taylor(resul-4, twos=0, 2*n+3) ; resul := coeftayl(resul, twos=0, 2*n) ; resul := resul*4^n/2 ; abs(resul*factorial(n+1)*factorial(n)) ; end: seq(A131490(n), n=1..23) ; # R. J. Mathar, Jul 31 2007
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Jul 28 2007
EXTENSIONS
More terms from R. J. Mathar, Jul 31 2007
STATUS
approved