A263646 Coefficients for an expansion of the Schwarzian derivative of a power series.
1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 5, 1, 1, 1, 6, 1, 1, 1, 7, 1, 1, 1, 1, 8, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1
Examples
Partitions by powers of x^n:
n=0: -(F2 + F1^2/2)
n=1: -(2 F3 + F1 F2)
n=2: -(3 F4 + F1 F3 + F2^2/2) = -[3 F4 + (F1 F3 + F2 F2 + F3 F1) / 2]
n=3: -(4 F5 + F1 F4 + F2 F3) = -[4 F5 + (F1 F4 + F2 F3 + F3 F2 + F4 F1) / 2]
n=4: -(5 F6 + F1 F5 + F2 F4 + F3^2/2)
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Example series:
f(x)= (1/2) / (1-x)^2 = 1/2 + x + (3/2) x^2 + 2x^3 + (5/2)x^4 + ... .
log(f'(x)) = log(1 + 3x + 6x^2 + 10x^3 + ...) = 3x + 3 x^2/2 + 3 x^3/3 + ... .
Then F(n) = -3 for n>=1, and the Schwarzian derivative series is
S{f(x)} = - [(-3 + 3^2/2) + (-2*3 + 3^2) x + (-3*3 + 3^2 + 3^2/2) x^2 + ...] = -3/2 - 3x - (9/2)x^2 - 6x^3 - (15/2)x^4 - ... .
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The Schwarzian vanishes if and only if acting on a Moebius, or linear fractional, transformation. This corresponds to F(n) = (-1)^(n+1) 2 * d^n, where d is an arbitrary constant.
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Example polynomial:
f(x) = (x-x1)(x-x2)/-(x1+x2)
log(f'(x)) = log[1 - 2x/(x1+x2)] = Sum_{n>= 1} -(2/(x1+x2))^n x^n/n.
Then F(n) = (2/(x1+x2))^n, and the Schwarzian derivative series is
S{f(x)} = (Sum_{n>= 0} -6(n+1) 2^n (x/(x1+x2))^n) / (x1+x2)^2 = -6 / (x1+x2-2*x)^2 (cf. A001787 and A085750).
Programs
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Python
print(sum(([n]+[1]*((n+1)//2) for n in range(1, 18)), [])) # Andrey Zabolotskiy, Mar 07 2024
Formula
Schwarzian{f(x)} = S{f(x)} = (D^3 f(x)) / (D f(x)) - (3/2) [(D^2 f(x)) / D f(x)]^2 = D [(D^2 f(x)) / D f(x)] - (1/2) [(D^2 f(x)) / D f(x)]^2 = D^2 log[D f(x)] - (1/2) [D log[D f(x)]]^2.
Then, with G(x) = log[D f(x)], S{f(x)} = D^2 G(x) - (1/2) [D G(x)]^2.
With f'(0) = 1, G(x) = log[D f(x)] = sum[n >= 1, -F(n) * x^n/n], and F(n) as Fn,
S{f(x)} = -[(F2 + F1^2/2) + (2 F3 + F1 F2) x + (3 F4 + F1 F3 + F2^2/2) x^2 + (4 F5 + F1 F4 + F2 F3) x^3 + (5 F6 + F1 F5 + F2 F4 + F3^2/2) x^4 + (6 F7 + F1 F6 + F2 F5 + F3 F4) x^5 + (7 F8 + F1 F7 + F2 F6 + F3 F5 + F4^2/2) x^6 + ...] .
This entry's a(m) are the numerators of the coefficients of the binary partitions in the brackets. For the singular partition of the integer n, the coefficient is (n-1); for the symmetric partition, 1/2; and for the rest, 1.
More symmetrically, x^2 S{f(x)}= - sum{n>=2, x^n [(n-1)F(n) + (1/2) sum(k=1 to n-1, F(n-k) F(k))]}.
With f(x)= c(0) + x + c(2) x^2 + ... , F(n) are given by the Faber polynomials of A263916: F(n) = Faber(n,2c(2),3c(3),..,(n+1)c(n+1)).
Extensions
More terms from Tom Copeland, Oct 01 2016
Comments