A030266 Shifts left under COMPOSE transform with itself.

0, 1, 1, 2, 6, 23, 104, 531, 2982, 18109, 117545, 808764, 5862253, 44553224, 353713232, 2924697019, 25124481690, 223768976093, 2062614190733, 19646231085928, 193102738376890, 1956191484175505, 20401540100814142, 218825717967033373, 2411606083999341827

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..220

David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

David Callan, Lagrange Inversion Counts 3bar-5241-Avoiding Permutations, J. Int. Seq. 14 (2011) # 11.9.4.

N. J. A. Sloane, Transforms

Formula

G.f. A(x) satisfies the functional equation: A(x)-x = x*A(A(x)). - Paul D. Hanna, Aug 04 2002

G.f.: A(x/(1+A(x))) = x. - Paul D. Hanna, Dec 04 2003

Suppose the functions A=A(x), B=B(x), C=C(x), etc., satisfy: A = 1 + xAB, B = 1 + xABC, C = 1 + xABCD, D = 1 + xABCDE, etc., then B(x)=A(x*A(x)), C(x)=B(x*A(x)), D(x)=C(x*A(x)), etc., where A(x) = 1 + x*A(x)*A(x*A(x)) and x*A(x) is the g.f. of this sequence (see table A128325). - Paul D. Hanna, Mar 10 2007

G.f. A(x) = x*F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1)*(1 + x*F(x,n+1)) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007

a(n) = [x^(n-1)] [1 + A(x)]^n/n for n>=1 with a(0)=0; i.e., a(n) equals the coefficient of x^(n-1) in [1+A(x)]^n/n for n >= 1. - Paul D. Hanna, Nov 18 2008

From Paul D. Hanna, Jul 09 2009: (Start)

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then

a(n,m) = Sum_{k=0..n} m*C(n+m,k)/(n+m) * a(n-k,k).

(End)

G.f. satisfies:

* A(x) = x*exp( Sum_{m>=0} {d^m/dx^m A(x)^(m+1)/x} * x^(m+1)/(m+1)! );

* A(x) = x*exp( Sum_{m>=1} [Sum_{k>=0} C(m+k-1,k)*{[y^k] A(y)^m/y^m}*x^k]*x^m/m );

which are equivalent. - Paul D. Hanna, Dec 15 2010

The g.f. satisfies:

log(A(x)/x) = A(x) + {d/dx A(x)^2/x}*x^2/2! + {d^2/dx^2 A(x)^3/x}*x^3/3! + {d^3/dx^3 A(x)^4/x}*x^4/4! + ... - Paul D. Hanna, Dec 15 2010

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + ...
A(A(x)) = x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + ...

Maple

A:= proc(n) option remember;
      unapply(`if`(n=0, x,
      A(n-1)(x)+coeff(A(n-1)(A(n-1)(x)), x, n) *x^(n+1)), x)
    end:
a:= n-> coeff(A(n)(x), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 24 2012

Mathematica

A[0] = Identity; A[n_] := A[n] = Function[x, Evaluate[A[n-1][x]+Coefficient[A[n-1][A[n-1][x]], x, n]*x^(n+1)]]; a[n_] := Coefficient[A[n][x], x, n]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
CoefficientList[Nest[x + x (# /. x -> #) &, O[x], 30], x] (* Vladimir Reshetnikov, Aug 08 2019 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+x*A*subst(A, x, x*A+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Mar 10 2007
(PARI) {a(n)=local(A=sum(i=1, n-1, a(i)*x^i)+x*O(x^n)); if(n==0, 0, polcoeff((1+A)^n/n, n-1))} \\ Paul D. Hanna, Nov 18 2008
(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n+m, k)/(n+m)*a(n-k, k))))} \\ Paul D. Hanna, Jul 09 2009
(PARI) {a(n)=local(A=1+sum(i=1, n-1, a(i)*x^i+x*O(x^n)));
for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n-m, binomial(m+k-1, k)*polcoeff(A^(2*m), k)*x^k)*x^m/m)+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Dec 15 2010

Crossrefs

Cf. A001028, A035049.

Cf. A110447.

Cf. A128325, A121687.

Sequence in context: A117106 A137534 A137535 * A110447 A137536 A137537

Adjacent sequences: A030263 A030264 A030265 * A030267 A030268 A030269

Keyword

nonn,nice,eigen

Author

Christian G. Bower

Status

approved