A087949 G.f. satisfies A(x) = 1 + x*A(x*A(x)).
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
Offset: 0
Keywords
Examples
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).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..250
Crossrefs
Programs
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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-François Alcover, Oct 04 2016, adapted from PARI *) -
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
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
Extensions
Edited by N. J. A. Sloane, May 19 2008