0,4

If offset is changed to 1, the following staetements hold: G.f. satisfies: A(x) = Series_Reversion[ 2*x/(1 + sqrt(1 + 4*A(x))) ]. Also, A(x) = x*(1 + sqrt(1 + 4*A(A(x))))/2 and A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 16*x^6 + 59*x^7 +246*x^8 +...; A(A(x)) = x + 2*x^2 + 4*x^3 + 10*x^4 + 30*x^5 + 105*x^6 + 416*x^7 +...; x = A(x*[1 - A(x) + 2*A(x)^2 - 5*A(x)^3 + 14*A(x)^4 - 42*A(x)^5 +-...]). - Paul D. Hanna (pauldhanna(AT)juno.com), May 15 2008

Comment from Paul D. Hanna (pauldhanna(AT)juno.com), Apr 16 2007: 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 ; ...

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)

(PARI) {a(n)=local(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))))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009]

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

Sequence in context: A000753 A007878 A019589 this_sequence A028333 A007747 A107283

Adjacent sequences: A087946 A087947 A087948 this_sequence A087950 A087951 A087952

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2003

Edited by N. J. A. Sloane (njas(AT)research.att.com), May 19 2008