A005119 Infinitesimal generator of x*(x + 1).

1, 1, 3, 16, 124, 1256, 15576, 226248, 3729216, 68179968, 1361836800, 29501349120, 693638208000, 17815908096000, 502048890201600, 15388268595840000, 500579319427891200, 16817771937344716800, 581609175119297740800

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• sign
• nice

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

Examples

			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)
		

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
• Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See pp. 21, 24.
• Gilbert Labelle, Sur l'Inversion et l'Iteration Continue des Séries Formelles, European Journal of Combinatorics, Vol. 1 Issue 2 (June 1980), 113-138.

Crossrefs

Cf. A030528.

Programs

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 *)

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

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

Extensions

More terms from Paul D. Hanna, Dec 27 2007