The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A030266 Shifts left under COMPOSE transform with itself. 40
 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 (list; graph; refs; listen; history; text; internal format)
 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: A137534 A137535 A110447 * A137536 A137537 A137538 Adjacent sequences: A030263 A030264 A030265 * A030267 A030268 A030269 KEYWORD nonn,nice,eigen AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 9 05:05 EST 2023. Contains 360153 sequences. (Running on oeis4.)