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

1,3

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

Table of n, a(n) for n=1..19.

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

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

Cf. A001263, A103364.

Sequence in context: A218827 A120021 A223897 * A121673 A273997 A051921

Adjacent sequences: A131487 A131488 A131489 * A131491 A131492 A131493

Keyword

nonn

Author

Jonathan Vos Post, Jul 28 2007

Extensions

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

Status

approved