A247331 - OEIS

%I #3 Sep 14 2014 21:34:36

%S 1,4,20,148,1492,18068,244628,3582612,55783252,913716116,15633525524,

%T 278068128660,5124595687636,97633146977428,1919960496128660,

%U 38930551809036436,813367272118600276,17501331733030883732,387693438148021391892,8839040069648710445460

%N  G.f. satisfies: A(x) = Sum_{n>=0} x^n * (3 + A(x)^n)^n.

%F  G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2)/(1 - 3*x*A(x)^n)^(n+1).

%e  G.f.: A(x) = 1 + 4*x + 20*x^2 + 148*x^3 + 1492*x^4 + 18068*x^5 +...

%e where the g.f. satisfies following series identity:

%e A(x) = 1 + x*(3+A(x)) + x^2*(3+A(x)^2)^2 + x^3*(3+A(x)^3)^3 + x^4*(3+A(x)^4)^4 + x^5*(3+A(x)^5)^5 + x^6*(3+A(x)^6)^6 +...

%e A(x) = 1/(1-3*x) + x*A(x)/(1-3*x*A(x))^2 + x^2*A(x)^4/(1-3*x*A(x)^2)^3 + x^3*A(x)^9/(1-3*x*A(x)^3)^4 + x^4*A(x)^16/(1-3*x*A(x)^4)^5 + x^5*A(x)^25/(1-3*x*A(x)^5)^6 + x^6*A(x)^36/(1-3*x*A(x)^6)^7 +...

%o  (PARI) {a(n,t=3)=local(A=1+x); for(i=1, n, A=sum(k=0, n, A^(k^2)*x^k/(1 - t*A^k*x +x*O(x^n))^(k+1) )); polcoeff(A, n)}

%o for(n=0,25,print1(a(n),", "))

%o (PARI) {a(n,t=3)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^k * (t + A^k +x*O(x^n))^k)); polcoeff(A, n)}

%o for(n=0,25,print1(a(n),", "))

%Y  Cf. A247330, A203000.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 14 2014