OFFSET
1,3
COMMENTS
Coefficients for partition polynomials for compositional inversion order-by-order of odd functions, e.g.f.s, or formal Taylor series f(x) = a1 x + a3 x^3/3! + a5 x^5/5! + ... .
The compositional inverse of f(x) is g(x)
= a1^(-1) [1] x
+ a1^(-4) [-1 a3] x^3/3!
+ a1^(-7) [10 a3^2 - 1 a1 a5] x^5/5!
+ a1^(-10)[-280 a3^3 + 56 a1 a3 a5 - a1^2 a7] x^7/7!
+ a1^(-13)[15400 a3^4 - 4620 a1 a3^2 a5 + a1^2 (126 a5^2 + 120 a3 a7) - a1^3 a9] * x^9/9! ... .
MATHEMATICA
rows[nn_] := With[{s = InverseSeries[x + Sum[a[k] x^(2k+1)/(2k+1)!, {k, nn}] + O[x]^(2nn+2)]}, Table[(2n-1)! Coefficient[s, x^(2n-1) Product[a[w], {w, p}]], {n, nn}, {p, Reverse[Sort[Sort /@ IntegerPartitions[n-1]]]}]];
rows[5] // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)
CROSSREFS
KEYWORD
sign,easy,tabf
AUTHOR
Tom Copeland, Oct 12 2016
EXTENSIONS
Corrected and extended by Andrey Zabolotskiy, Mar 07 2024
STATUS
approved