A247330 - OEIS

%I #5 Sep 27 2014 14:26:53

%S 1,3,12,75,633,6330,70410,845490,10778385,144342129,2016502329,

%T 29249703273,439097183598,6807064047249,108811265375748,

%U 1791748638013341,30373586425246566,529855701281428431,9509268033398381151,175539561089425403601,3332349856995500161920,65037265540406292591147

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * (2 + A(x)^n)^n.

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

%e G.f.: A(x) = 1 + 3*x + 12*x^2 + 75*x^3 + 633*x^4 + 6330*x^5 + 70410*x^6 +...

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

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

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

%o (PARI) {a(n,t=2)=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=2)=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. A203000, A247331.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 13 2014