OFFSET
0,2
COMMENTS
The sequence results from some interesting combinatorics that come from the interaction of
1) the quotient rule,
2) the derivative recurrence relation for Bessel functions, and
3) the fact that J_n(0) = 0 unless n = 0 and J_0(0) = 1.
At each stage the numerator is the sum of products of Bessel functions. Each derivative introduces new terms which also (by the Bessel derivative recurrence relation) increment or decrement the Bessel function orders in the terms of the numerator. a(n) is the coefficient in front of the term (J_0(x))^(2^n) (if it exists). There are complicated combinatorics involved in what contributes to that coefficient.
LINKS
Christoph Aistleitner, Zakhar Kabluchko, and Joscha Prochno, Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits, arXiv:2512.15501 [math.NT], 2025. See p. 3.
Eric Weisstein's World of Mathematics, Bessel Function of the First Kind.
FORMULA
a(n) = -2^(2n)*(d^(2n)/dx^(2n))(log(J_0(x)))|_x=0.
An expression which does not require derivatives is:
a(n) = 2^(2n)*Sum_{k=1..2n} (-1)^k*(k-1)!*Y_{2n,k}(d_i),
where Y_{n,k} is the partial Bell polynomial with inputs given by
d_{2i} = (-1)^i*2^(-2i)*binomial(2i,i) and d_{2i+1} = 0.
EXAMPLE
For n = 0:
-1*(d^0/dx^0)log(J_0(x)) = -log(J_0(0)) = 0.
For n = 1:
-4*(d^2/dx^2)log(J_0(x)) = (-J_{-2}J_0 + 2J_0^2 - J_2J_0 + J_{-1}J_{-1} -
2J_{-1}J_1 + J_1 J_1)/(J_0)^4, which evaluates to 2 (because of 2J_0^2).
MAPLE
a := n -> -2^(2*n)*limit(diff(log(BesselJ(0, x)), [x$2*n]), x=0):
seq(a(n), n=0..5);
MATHEMATICA
Table[2^n*Sum[(-1)^k*Factorial[k - 1]*BellY[n, k, Flatten[Table[{0, (-1)^{(j + 1)/2}*2^{-(j + 1)}*Binomial[j + 1, (j + 1)/2]}, {j, 1, N - k + 1, 2}]]], {k, 1, n}], {n, 0, 32, 2}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Geoff Goehle, Mar 11 2022
STATUS
approved