A120955 G.f. A(x) satisfies: Series_Reversion( x/A(x) ) / x = 2*A(x) - (1+x).

1, 1, 1, 4, 25, 206, 2060, 23920, 314065, 4582300, 73393490, 1278859176, 24073541260, 486806278752, 10525038917720, 242318610557760, 5919811842140945, 152974724047702626, 4169576527021400852

Offset

0,4

Comments

The g.f. for A120956 = Series_Reversion( x/A(x) ) / x = 2*A(x) - (1+x), where A(x) is the g.f. of this sequence.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..256

Formula

G.f. A(x) satisfies:

(1) A( x*(2*A(x) - 1-x) ) = 2*A(x) - 1-x.

(2) A(x) = 2*A(x/A(x)) - 1 - x/A(x).

(3) A(x) = F(x/A(x)) and F(x) = A(x*F(x)) where F(x) = g.f. of A120956.

(4) A(x) = 1+x + Sum{n>=1} x^n * d^n/dx^n (A(x)-1)^(n+1) / (n+1)!. - Paul D. Hanna, Dec 27 2012

a(n) = A120956(n)/2 for n>=2.

a(n) = [x^n] A(x)^n / (2*n) for n>1; i.e., a(n) equals the coefficient of x^n in A(x)^n divided by 2*n.

a(n) ~ c * n^(n + 1/2 + log(2)) / (exp(n) * (log(2))^n), where c = 0.33794865962155... . - Vaclav Kotesovec, Aug 10 2014

Example

A(x) = 1 + x + x^2 + 4*x^3 + 25*x^4 + 206*x^5 + 2060*x^6 +...
The g.f. of A120956 is:
series_reversion(x/A(x))/x = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 412*x^5 +...
Compare terms to see that A120956(n) = 2*a(n) for n>=2.
The g.f. satisfies the series:
A(x) = 1+x + x*d/dx (A(x)-1)^2/2! + x^2*d^2/dx^2 (A(x)-1)^3/3! + x^3*d^3/dx^3 (A(x)-1)^4/4! + x^4*d^4/dx^4 (A(x)-1)^5/5! +...

Prog

(PARI) a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, t); A[ #A]=subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); A[n+1]
for(n=0, 25, print1(a(n), ", "))
(PARI) Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D
{a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=1+x+sum(m=1, n, x^m*Dx(m, (A-1+x*O(x^n))^(m+1)/(m+1)!) )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 27 2012

Crossrefs

Cf. A120956, A120970 (variant), A249933.

Sequence in context: A301363 A213192 A036242 * A061714 A005411 A105628

Adjacent sequences: A120952 A120953 A120954 * A120956 A120957 A120958

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2006

Status

approved