0,4

The formula for the inversion of the power series y=F(x)= x*G(x)= x*(1 + sum(g[k]*(x^k),k=1..infinity)) is obtained as a corollary of Lagrange's inversion theorem. The result is F^{(-1)}(y)= sum(P(n-1)*y^n,n=1..infinity), where P(n):=sum over partitions of n of a(n,k)* G[k], with G[k]:=g[1]^e(k,1)*g[2]^e(k,2)*...*g[n]^e(k,n) if the k-th partition of n, in Abramowitz-Stegun order(see the given ref, pp. 831-2), is [1^e(k,1),2^e(k,2),...,n^e(k,n)], for k=1..p(n):= A000041(n) (partititon numbers).

The sequence of row lengths is A000041(n) (partition numbers).

The signs are given by (-1)^m(n,k), with the number of parts m(n,k) = sum(e(k,j),j=1..n) of the k-th partition of n. For m(n,k) see A036043.

The proof that the unsigned row sums give Schroeder's little numbers A001003(n) results from their formula (diff(((1-x)/(1-2*x))^n,x$(n-1)))/n!|_{x=0}, n>=1. This formula for A001003 can be proved starting with the compositional inverse of the g.f. of A001003 (which is given there in a comment) and using Lagrange's inversion theorem to recover the original sequence A001003.

For alternate formulations and relation to the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 150, Table 4.1 (unsigned).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16, 3.6.25.

W. Lang, First 10 rows and a formula.

Eric Weisstein's MathWorld Series Reversion.

For row n>=1 the row polynomial in the variables g[1], ..., g[n] is P(n)=(1/(n+1)!)*diff(1/G(x)^(n+1), x$n)|_{x=0}. P(0):=1. diff(G(x), x$k)|_{x=0} = k!*g[k], k>=1; G(0)=1.

a(n, k) is the coefficient in P(n) of g[1]^e(k, 1)*g[2]^e(k, 2)*..*g[n]^e(k, n) with the k-th partition of n written as [1^e(k, 1), 2^e(k, 2), ..., n^e(k, n)] in Abramowitz-Stegun order (e(k, j)>=0. If e(k, j)=0 then j^0 is not recorded).

[1]; [ -1]; [ -1,2]; [ -1,5,-5];

[ -1,6,3,-21,14];[ -1,7,7,-28,-28,84,-42]; ...

[ -1,6,3,-21,14], the fifth row, stands for the row polynomial P(4) = -1*g[4] + 6*g[1]*g[3] + 3*g[2]^2 - 21*(g[1]^2)*g[2] + 14*g[1]^4 = (1/5!)*(differentiate 1/G(x)^5 four times and evaluate at x = 0). This gives the coefficient of y^5 of F^{(-1)}(y).

Row sums give (-1)^n. Unsigned row sums are A001003(n) (little Schroeder numbers).

Sequence in context: A141485 A005605 A145882 this_sequence A021468 A033282 A126350

Adjacent sequences: A111782 A111783 A111784 this_sequence A111786 A111787 A111788

sign,tabf

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005