%I A087949
%S A087949 1,1,1,2,5,16,59,246,1131,5655,30428,174835,1066334,6870542,46581883,
%T A087949 331237074,2463361903,19112314727,154364077009,1295325828045,
%U A087949 11273167827343,101589943242179,946577526626181,9107029927925714
%N A087949 G.f. satisfies A(x) = 1 + xA(xA(x)).
%F A087949 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
%F A087949 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:
%F A087949 A = 1 + xB;
%F A087949 B = 1 + xAC;
%F A087949 C = 1 + xABD;
%F A087949 D = 1 + xABCE;
%F A087949 E = 1 + xABCDF ; ...
%F A087949 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 09 2009: 
               (Start)
%F A087949 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n, then
%F A087949 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.
%F A087949 (End)
%o A087949 (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))}
%o A087949 (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]
%Y A087949 Cf. A002449, A030266, A088714, A088717, A091713, A120971, A140092, A000108.
%Y A087949 Sequence in context: A000753 A007878 A019589 this_sequence A028333 A007747 
               A107283
%Y A087949 Adjacent sequences: A087946 A087947 A087948 this_sequence A087950 A087951 
               A087952
%K A087949 nonn
%O A087949 0,4
%A A087949 Paul D. Hanna (pauldhanna(AT)juno.com), Sep 16 2003
%E A087949 Edited by N. J. A. Sloane (njas(AT)research.att.com), May 19 2008