A002474 Denominators of coefficients of odd powers of x of the expansion of Bessel function J_1(x).

2, 16, 384, 18432, 1474560, 176947200, 29727129600, 6658877030400, 1917756584755200, 690392370511872000, 303772643025223680000, 160391955517318103040000, 100084580242806496296960000, 72861574416763129304186880000, 61203722510081028615516979200000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The corresponding numerators are A033999(n) = (-1)^n.

Examples

			a(3) = 18432 = 128*6*24, since J_{1}(x) = x/2 - x^3/16 + x^5/384 - x^7/18432 + ...
		

References

• Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.4.7
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:2 and 52:6:3 at page 483, 513.

Links

• T. D. Noe, Table of n, a(n) for n = 0..50
• Index to divisibility sequences.
• Index entries for sequences related to Bessel functions or polynomials.

Crossrefs

Cf. J_0: A002454, J_2: A002506, J_3: A014401, J_4: A061403, J_5: A061404, J_6: A061405, J_7: A061407, J_9: A061440 J_10: A061441.

Cf. A010790, A033999.

Programs

Magma

[2^(2*n+1)*Factorial(n)*Factorial(n+1): n in [0..30]]; // G. C. Greubel, Sep 21 2024

Maple

a:= n-> denom(coeff(series(BesselJ(1, x), x, 2*n+2), x, 2*n+1)):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 21 2024

Mathematica

CoefficientList[Series[BesselJ[1,x], {x,0,30}], x][[2 ;; ;; 2]]//Denominator
Table[2^(2*n+1)*n!*(n+1)!, {n,0,30}] (* G. C. Greubel, Sep 21 2024 *)

PARI

a(n) = n!^2 * (n+1) << (2*n+1) \\ Charles R Greathouse IV, Oct 23 2023

PARI

first(n)=my(x='x+O('x^(2*n+1)),t=besselj(1,x)); vector(n+1,k,2*denominator(polcoeff(t,2*k-2))) \\ Charles R Greathouse IV, Oct 23 2023

SageMath

[2^(2*n+1)*factorial(n)*factorial(n+1) for n in range(31)] # G. C. Greubel, Sep 21 2024

Formula

a(n) = 2^(2n+k) * n! * (n+k)! here for k=1, i.e., Bessel's J1(x) has the denominator a(n) for the coefficient of x^(2*n+1), n >= 0.

a(n) = 2^(2n+1)*A010790(n).

Extensions

Name specified, numerators given, formula augmented by Wolfdieter Lang, Aug 25 2015