%I A111785
%S A111785 1,1,1,2,1,5,5,1,6,3,21,14,1,7,7,28,28,84,42,1,8,8,4,36,72,12,120,180,
%T A111785 330,132,1,9,9,9,45,90,45,45,165,495,165,495,990,1287,429,1,10,10,10,5,
               55,
%U A111785 110,110,55,55,220,660,330,660,55,715,2860,1430,2002,5005,5005,1430,1,
               11,11
%V A111785 1,-1,-1,2,-1,5,-5,-1,6,3,-21,14,-1,7,7,-28,-28,84,-42,-1,8,8,4,-36,-72,
               -12,120,180,
%W A111785 -330,132,-1,9,9,9,-45,-90,-45,-45,165,495,165,-495,-990,1287,-429,-1,
               10,10,10,5,-55,
%X A111785 -110,-110,-55,-55,220,660,330,660,55,-715,-2860,-1430,2002,5005,-5005,
               1430,-1,11,11
%N A111785 Array used for power series inversion (sometimes called reversion).
%C A111785 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).
%C A111785 The sequence of row lengths is A000041(n) (partition numbers).
%C A111785 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.
%C A111785 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$(n-1)))/
               n!|_{x=0}, n>=1. This formula for A001003 can be proved starting 
               with the compositional inverse of the g.f. of A001003 (which is given 
               there in a comment) and using Lagrange's inversion theorem to recover 
               the original sequence A001003.
%C A111785 For alternate formulations and relation to the geometry of associahedra 
               or Stasheff polytopes (and other combinatorial objects) see A133437. 
               [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]
%D A111785 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 150, Table 4.1 
               (unsigned).
%H A111785 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A111785 A. M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 
               1972, p. 16, 3.6.25.
%H A111785 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A111785.text">
               First 10 rows and a formula.</a>
%H A111785 Eric Weisstein's MathWorld <a href="http://mathworld.wolfram.com/SeriesReversion.html">
               Series Reversion.</a>
%F A111785 For row n>=1 the row polynomial in the variables g[1], ..., g[n] is P(n)=(1/
               (n+1)!)*diff(1/G(x)^(n+1), x$n)|_{x=0}. P(0):=1. diff(G(x), x$k)|_{x=0} 
               = k!*g[k], k>=1; G(0)=1.
%F A111785 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).
%e A111785 [1]; [ -1]; [ -1,2]; [ -1,5,-5];
%e A111785 [ -1,6,3,-21,14];[ -1,7,7,-28,-28,84,-42]; ...
%e A111785 [ -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).
%Y A111785 Row sums give (-1)^n. Unsigned row sums are A001003(n) (little Schroeder 
               numbers).
%Y A111785 Sequence in context: A141485 A005605 A145882 this_sequence A021468 A033282 
               A126350
%Y A111785 Adjacent sequences: A111782 A111783 A111784 this_sequence A111786 A111787 
               A111788
%K A111785 sign,tabf
%O A111785 0,4
%A A111785 Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 
               Aug 23 2005