|
%I #2 Mar 30 2012 18:37:08
%S 1,4,36,465,7552,145144,3191724,78633536,2139797572,63661720934,
%T 2054750445056,71506836054768,2669607994059332,106461335627737280,
%U 4518074113214973200,203369587888867105097,9680278850211643305728
%N G.f.: A(x) = 1 + x*[A_2(x)]^2, where A_2(x) = 2 + x*[A_3(x)]^2, A_3(x) = 3 + x*[A_4(x)]^2, ..., A_n(x) = n + x*[A_{n+1}(x)]^2, ...
%e G.f. A(x) = 1 + x*(2 + x*(3 + x*(4 + x*(5 +...)^2 )^2 )^2 )^2.
%e Initial nested functions, A_n(x) = n + x*[A_{n+1}(x)]^2, begin:
%e A(x) = 1 + 4x + 36x^2 + 465x^3 + 7552x^4 + 145144x^5 +...
%e A_2(x) = 2 + 9x + 96x^2 + 1456x^3 + 27430x^4 + 604608x^5 +...
%e A_3(x) = 3 + 16x + 200x^2 + 3505x^3 + 75408x^4 + 1879328x^5 +...
%e A_4(x) = 4 + 25x + 360x^2 + 7176x^3 + 173866x^4 + 4839640x^5 +...
%e A_5(x) = 5 + 36x + 588x^2 + 13153x^3 + 354688x^4 + 10913640x^5 +...
%e A_6(x) = 6 + 49x + 896x^2 + 22240x^3 + 660942x^4 + 22285664x^5 +...;
%e In general, A_n(x) = n + (n+1)^2*x + 2(n+1)(n+2)^2*x^2 + [4(n+1)(n+2)(n+3)^2+(n+2)^4]x^3 + [8(n+1)(n+2)(n+3)(n+4)^2+2(n+1)(n+3)^4+4(n+2)^3(n+3)^2]x^4 +...
%o (PARI) {a(n)=local(A=n+1+x*(n+2)^2);if(n==0,1,for(i=0,n-1,A=(n-i+x*(A+x*O(x^n))^2)); polcoeff(A,n))}
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 05 2007
| 