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 modified Bessel functions, and
3) the fact that I_n(0) = 0 unless n = 0 and I_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 (I_0(x))^(2^n) (if it exists). There are complicated combinatorics involved in what contributes to that coefficient.
LINKS
Eric Weisstein's World of Mathematics, Modified Bessel Functions of the First Kind
FORMULA
a(n) = 2^(2n)*(d^(2n)/dx^(2n))(log(I_0(x)))|_x=0.
An expression which does not require derivatives is:
a(n) = 2^(2n)*Sum_{k=1..2n} (-1)^(k-1)(k-1)!*Y_{2n,k}(d_i), where Y_{n,k} is the partial Bell polynomial with inputs given by d_{2i} = 2^(-2i)*binomial(2i,i) and d_{2i+1} = 0.
EXAMPLE
For n = 0:
1*(d^0/dx^0)log(I_0(x)) = log(I_0(0)) = 0.
For n = 1:
4*(d^2/dx^2)log(I_0(x)) = (I_{-2}I_0 + 2I_0^2 + I_2I_0 - I_{-1}I_{-1} -
2I_{-1}I_1 + I_1 I_1)/(I_0)^4 which evaluates to 2 (because of the 2I_0^2).
MAPLE
A := proc(n) option remember; ifelse(n=2, (-BesselI(1, x)^2*x + BesselI(0, x)^2*x - BesselI(1, x)* BesselI(0, x))/(BesselI(0, x)^2*x), simplify(diff(A(n-1), x))) end:
a := n -> ifelse(n=0, 0, 2^n*limit(A(n), x=0)):
seq(a(n), n=0..32, 2); # Peter Luschny, Mar 11 2022
MATHEMATICA
Table[2^n*Sum[(-1)^(k-1)*Factorial[k-1]*BellY[n, k, Flatten[Table[{0, 2^{-(j + 1)}* Binomial[j+1, (j+1)/2]}, {j, 1, n-k+1, 2}]]], {k, 1, n}], {n, 0, 32, 2}]
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Geoff Goehle, Mar 10 2022
STATUS
approved