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
Examples
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 + ...
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
Programs
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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 *) -
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
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
Comments