A277394 Lagrange inversion, or reversion, for divided power series with odd powers only.

1, -1, 10, -1, -280, 56, -1, 15400, -4620, 126, 120, -1, -1401400, 560560, -36036, -17160, 792, 220, -1, 190590400, -95295200, 10090080, 3203200, -126126, -360360, -50050, 1716, 2002, 364, -1

Offset: 1

Author

Tom Copeland, Oct 12 2016

Keywords

• sign
• easy
• tabf

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! ... .

Crossrefs

Cf. A133437, A134264, A134685, A133932, A145271, A176740 for other inversion formulas.

Programs

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 *)

Extensions

Corrected and extended by Andrey Zabolotskiy, Mar 07 2024