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A087949 G.f. satisfies A(x) = 1 + x*A(x*A(x)). 12
1, 1, 1, 2, 5, 16, 59, 246, 1131, 5655, 30428, 174835, 1066334, 6870542, 46581883, 331237074, 2463361903, 19112314727, 154364077009, 1295325828045, 11273167827343, 101589943242179, 946577526626181, 9107029927925714, 90359115887726302, 923509462029444933 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..250

FORMULA

Let G(x) = x*A(x), then the following statements hold:

* G(x) = x*(1 + sqrt(1 + 4*G(G(x))))/2;

* G(x) = Series_Reversion[2*x/(1 + sqrt(1 + 4*G(x)))].

- Paul D. Hanna, May 15 2008

From Paul D. Hanna, Apr 16 2007: (Start)

G.f. A(x) is the unique solution to variable A in the infinite system of simultaneous equations:

A = 1 + xB;

B = 1 + xAC;

C = 1 + xABD;

D = 1 + xABCE;

E = 1 + xABCDF ; ... (End)

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

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

a(n,m) = Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * a(n-k,k) with a(0,m)=1.

(End)

G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)] *A(x)^(-2n-2)/(n+1)! ). - Paul D. Hanna, Dec 18 2010

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 +...

A(xA(x)) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 59*x^5 +...

Logarithmic series:

log(A(x)) = x/A(x) + [d/dx x^3*A(x)^2]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^3]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^4]*A(x)^(-8)/4! +...

Let G(x) = x*A(x) then

x = G(x*[1 - G(x) + 2*G(x)^2 - 5*G(x)^3 + 14*G(x)^4 - 42*G(x)^5 +-...])

where the unsigned coefficients are the Catalan numbers (A000108).

MAPLE

A:= proc(n) option remember; `if`(n=0, 1, (T->

      unapply(convert(series(1+x*T(x*T(x)), x, n+1)

      , polynom), x))(A(n-1)))

    end:

a:= n-> coeff(A(n)(x), x, n):

seq(a(n), n=0..25);  # Alois P. Heinz, May 15 2016

MATHEMATICA

a[n_] := (A=x; If[n<1, 0, For[i=1, i <= n, i++, A = InverseSeries[2*(x/(1 + Sqrt[1 + 4*A + x*O[x]^n]))]]]; SeriesCoefficient[A, {x, 0, n}]); Array[a, 26] (* Jean-Fran├žois Alcover, Oct 04 2016, adapted from PARI *)

PROG

(PARI) {a(n)=my(A=x); if(n<1, 0, for(i=1, n, A=serreverse(2*x/(1 + sqrt(1+4*A +x*O(x^n))))); polcoeff(A, n))}

(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, m*binomial(n-k+m, k)/(n-k+m)*a(n-k, k))))} \\ Paul D. Hanna, Jul 09 2009

(PARI) /* n-th Derivative: */

{Dx(n, F)=my(D=F); for(i=1, n, D=deriv(D)); D}

/* G.f. */

{a(n)=my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=0, n, Dx(m, x^(2*m+1)*A^(m+1))*A^(-2*m-2)/(m+1)!)+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Dec 18 2010

CROSSREFS

Cf. A002449, A030266, A088714, A088717, A091713, A120971, A140092, A000108.

Cf. A139702, A143426, A143435, A182969.

Sequence in context: A000753 A007878 A019589 * A028333 A007747 A208988

Adjacent sequences:  A087946 A087947 A087948 * A087950 A087951 A087952

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 16 2003

EXTENSIONS

Edited by N. J. A. Sloane, May 19 2008

STATUS

approved

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Last modified October 23 21:37 EDT 2018. Contains 316541 sequences. (Running on oeis4.)