A002474 - OEIS

%I #31 Sep 21 2024 19:38:13

%S 2,16,384,18432,1474560,176947200,29727129600,6658877030400,

%T 1917756584755200,690392370511872000,303772643025223680000,

%U 160391955517318103040000,100084580242806496296960000,72861574416763129304186880000,61203722510081028615516979200000

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

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

%D Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.4.7

%H T. D. Noe, <a href="/A002474/b002474.txt">Table of n, a(n) for n = 0..50</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%F 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.

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

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

%p a:= n-> denom(coeff(series(BesselJ(1, x), x, 2*n+2), x, 2*n+1)):

%p seq(a(n), n=0..15);  # _Alois P. Heinz_, Sep 21 2024

%t CoefficientList[Series[BesselJ[1,x], {x,0,30}], x][[2 ;; ;; 2]]//Denominator

%t Table[2^(2*n+1)*n!*(n+1)!, {n,0,30}] (* _G. C. Greubel_, Sep 21 2024 *)

%o (PARI) a(n) = n!^2 * (n+1) << (2*n+1) \\ _Charles R Greathouse IV_, Oct 23 2023

%o (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

%o (Magma) [2^(2*n+1)*Factorial(n)*Factorial(n+1): n in [0..30]]; // _G. C. Greubel_, Sep 21 2024

%o (SageMath) [2^(2*n+1)*factorial(n)*factorial(n+1) for n in range(31)] # _G. C. Greubel_, Sep 21 2024

%Y 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.

%Y Cf. A010790, A033999.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

%E Name specified, numerators given, formula augmented by _Wolfdieter Lang_, Aug 25 2015