A005119 Infinitesimal generator of x*(x + 1). (Formerly M3024)

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

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

Cf. A030528.

Sequence in context: A035352 A159607 A087018 * A190291 A090135 A351423

Adjacent sequences: A005116 A005117 A005118 * A005120 A005121 A005122

Keyword

sign,nice

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Paul D. Hanna, Dec 27 2007

Status

approved