|
| |
|
|
A030266
|
|
Shifts left under COMPOSE transform with itself.
|
|
32
|
|
|
|
0, 1, 1, 2, 6, 23, 104, 531, 2982, 18109, 117545, 808764, 5862253, 44553224, 353713232, 2924697019, 25124481690, 223768976093, 2062614190733, 19646231085928, 193102738376890, 1956191484175505, 20401540100814142
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
REFERENCES
|
David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.
|
|
|
LINKS
|
Alois P. Heinz, Table of n, a(n) for n = 0..150
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. [From Paul D. Hanna, Nov 18 2008]
Contribution 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
|
max = 22; f[x_] = Sum[a[n]*x^n, {n, 0, max}]; a[0] = 0; a[1] = a[2] = 1; coes = CoefficientList[ Series[f[x] - x - x*f[f[x]], {x, 0, max}], x] ; sol = SolveAlways[ Thread[coes == 0], x] // First; Table[a[n] /. sol, {n, 0, max}](* From Jean-François Alcover, May 14 2012, after Paul D. Hanna *)
|
|
|
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))} [From 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))))} [From 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)} [From 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
|
Christian G. Bower
|
|
|
STATUS
|
approved
|
| |
|
|
