login
G.f. a(x) satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^A000265(n) where A000265(n) = largest odd divisor of n.
1

%I #6 Mar 30 2012 18:37:26

%S 1,1,2,4,10,25,68,193,565,1688,5136,15854,49517,156191,496836,1591924,

%T 5133091,16643856,54234349,177505376,583272256,1923482331,6363842492,

%U 21117432227,70265970878,234388421515,783664894313,2625748635300

%N G.f. a(x) satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^A000265(n) where A000265(n) = largest odd divisor of n.

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

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 68*x^6 + 193*x^7 +...

%e The g.f. satisfies the following identities:

%e A(x) = 1 + x*A(x) + x^2*A(x) + x^3*A(x)^3 + x^4*A(x) + x^5*A(x)^5 + x^6*A(x)^3 + x^7*A(x)^7 + x^8*A(x) +...+ x^n*A(x)^A000265(n) +...

%e A(x) = 1 + x*A(x)/(1-x^2*A(x)^2) + x^2*A(x)/(1-x^4*A(x)^2) + x^4*A(x)/(1-x^8*A(x)^2) + x^8*A(x)/(1-x^16*A(x)^2) +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(m/2^valuation(m,2))));polcoeff(A,n)}

%Y Cf. A191769.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 16 2011