cp's OEIS Frontend

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

%I A277394 #13 Mar 08 2024 01:36:55
%S A277394 1,-1,10,-1,-280,56,-1,15400,-4620,126,120,-1,-1401400,560560,-36036,
%T A277394 -17160,792,220,-1,190590400,-95295200,10090080,3203200,-126126,
%U A277394 -360360,-50050,1716,2002,364,-1
%N A277394 Lagrange inversion, or reversion, for divided power series with odd powers only.
%C A277394 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! + ... .
%C A277394 The compositional inverse of f(x) is g(x)
%C A277394 = a1^(-1) [1] x
%C A277394 + a1^(-4) [-1 a3] x^3/3!
%C A277394 + a1^(-7) [10 a3^2 - 1 a1 a5] x^5/5!
%C A277394 + a1^(-10)[-280 a3^3 + 56 a1 a3 a5 - a1^2 a7] x^7/7!
%C A277394 + a1^(-13)[15400 a3^4 - 4620 a1 a3^2 a5 + a1^2 (126 a5^2 + 120 a3 a7) - a1^3 a9] * x^9/9! ... .
%t A277394 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]]]}]];
%t A277394 rows[5] // Flatten (* _Andrey Zabolotskiy_, Mar 07 2024 *)
%Y A277394 Cf. A133437, A134264, A134685, A133932, A145271, A176740 for other inversion formulas.
%K A277394 sign,easy,tabf
%O A277394 1,3
%A A277394 _Tom Copeland_, Oct 12 2016
%E A277394 Corrected and extended by _Andrey Zabolotskiy_, Mar 07 2024