A131490 Appears in Taylor series of powers of generalized Bessel functions.

1, 1, 3, 16, 130, 1485, 22645, 444136, 10889676, 326345460, 11736144420, 498798542880, 24732729791484, 1415034219327729, 92523874454996985, 6856434802243346320, 571604206230905727880, 53259509403796625217288, 5513868420471764306104008

Offset: 1

Author

Jonathan Vos Post, Jul 28 2007

Keywords

• nonn

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.

Crossrefs

Cf. A001263, A103364.

Programs

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

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

Extensions

More terms from R. J. Mathar, Jul 31 2007