%I #2 Mar 30 2012 18:37:10
%S 1,1,4,16,78,420,2454,15297,100660,694022,4986128,37171749,286619290,
%T 2279866324,18668221560,157080129914,1356186583276,11999018622158,
%U 108672944038356,1006528378511868,9525454067974148,92037443236217412
%N G.f.: A(x) = 1 + x*A_1(x)^2; A_1(x) = (1+x) + x*A_2(x)^2; A_2(x) = (1+x)^2 + x*A_3(x)^2; ...; A_{n}(x) = (1+x)^n + x*A_{n+1}(x)^2 for n>=0 with A(x) = A_0(x).
%F G.f.: A(x) = 1 + x*A_1(x)^2 where A_1(x) = g.f. of A138295.
%e G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 78*x^4 + 420*x^5 + 2454*x^6 +...
%e Given A_{n}(x) = (1+x)^n + x*A_{n+1}(x)^2 for n>=0,
%e the initial coefficients of the functions A_{n} for n=0..8 are:
%e A_0 = [1, 1, 4, 16, 78, 420, 2454, 15297, 100660, 694022, ...];
%e A_1 = [1, 2, 6, 27, 138, 789, 4878, 32114, 222690, 1614412,...];
%e A_2 = [1, 3, 9, 42, 228, 1377, 8992, 62400, 455252, 3465728,...];
%e A_3 = [1, 4, 13, 62, 356, 2266, 15586, 113752, 871378, 6953751,...];
%e A_4 = [1, 5, 18, 88, 531, 3554, 25676, 196609, 1577930, 13174337,...];
%e A_5 = [1, 6, 24, 121, 763, 5356, 40536, 324882, 2725852, 23763583,...];
%e A_6 = [1, 7, 31, 162, 1063, 7805, 61731, 516648, 4522200, 41085199,...];
%e A_7 = [1, 8, 39, 212, 1443, 11053, 91151, 794909, 7244078, 68460164,...];
%e A_8 = [1, 9, 48, 272, 1916, 15272, 131046, 1188417, 11254609, 110444000,..].
%o (PARI) {a(n)=local(A=1);for(i=0,n,A=(1+x)^(n-i)+x*(A+x*O(x^n))^2);polcoeff(A,n)}
%Y Cf. A138295 (A_1).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 13 2008