OFFSET
1,3
COMMENTS
From Peter Bala, Dec 09 2015: (Start)
Given a formal power series f(x) = x + f_2*x^2 + f_3*x^3 + ... Labelle [Section 4, Proposition 4] shows there is a power series w(x) = w_2*x^2 + w_3*x^3 + w_4*x^4 + ..., called the infinitesimal generator of f, such that the n-fold composition f^(n)(x) = f o f o ... o f (n factors) of f(x) is given by the operator exp( n*w(x)*d/dx ) acting on x. This gives the expansion f^(n)(x) = x + n/1!*w(x) + n^2/2!*w(x)*w'(x) + .... Taking n = -1 gives an expansion for the series reversion of f(x).
Let R denote the Riordan array (f(x)/x, f(x)). Then the coefficients of the infinitesimal generator w(x) form the first column of the matrix logarithm log(R).
Here we take f(x) = x + x^2 and calculate w(x) = x^2*(1 - x + 3*x^2/2! - 16*x^3/3! + 124*x^4/4! - ...). The numerators of the coefficients give a signed version of the present sequence. See the example below. (End)
a(29) = -307081193389527408920486163460915200000 is the first negative term. Georg Fischer, Feb 15 2019
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..200
Gilbert Labelle, Sur l'Inversion et l'Iteration Continue des Séries Formelles, European Journal of Combinatorics, Vol. 1 Issue 2 (June 1980), 113-138.
FORMULA
a(n) = (n-2)!*Sum_{i=1..n-1} (-1)^(i+1)*C(n-i+1,i+1)*a(n-i)/(n-i-1)! for n>1 with a(1)=1. E.g.f. satisfies: A(x) = (1-x)^2/(1-2x)*A(x-x^2) where A(x) = Sum_{n>=0}a(n+1)*x^n/n! with offset so that A(0)=1. - Paul D. Hanna, Dec 27 2007
EXAMPLE
From Peter Bala, Dec 09 2015: (Start)
The Riordan array R = (1 + x, x*(1 + x)) is A030528.
log(R) begins
/ 0
| 1 0
| -1 1*2 0
| 3/2! -1*2 1*3 0
|-16/3! (3/2!)*2 -1*3 1*4 0
|124/4! (-16/3!)*2 (3/2!)*3 -1*4 1*5 0
|...
\
The first column begins [1, -1, 3/2!, -16/3! 124/4!, ...]. (End)
MATHEMATICA
max = 19; f[x_] := Sum[a[n+1]*x^n/n!, {n, 0, max}]; coes = CoefficientList[ Series[ f[x]-((1-x)^2/(1-2*x))*f[x-x^2], {x, 0, max}], x]; Array[a, max] /. Solve[a[1] == a[2] == 1 && Thread[coes == 0]][[1]] (* Jean-François Alcover, Nov 03 2011 *)
nmax=20; a = ConstantArray[0, nmax]; a[[1]]=1; Do[a[[n]] = (n-2)! *Sum[(-1)^(i+1)*Binomial[n-i+1, i+1]*a[[n-i]]/(n-i-1)!, {i, 1, n-1}], {n, 2, nmax}]; a (* Vaclav Kotesovec, Mar 12 2014 *)
PROG
(PARI) {a(n)=if(n<1, 0, if(n==1, 1, (n-2)!*sum(i=1, n-1, (-1)^(i+1)*binomial(n-i+1, i+1)*a(n-i)/(n-i-1)!)))} \\ Paul D. Hanna, Dec 27 2007
CROSSREFS
KEYWORD
sign,nice
AUTHOR
EXTENSIONS
More terms from Paul D. Hanna, Dec 27 2007
STATUS
approved