A300050 - OEIS

%I #16 Apr 06 2018 20:27:32

%S 1,1,4,28,250,2558,28594,340486,4255344,55300776,742732646,

%T 10267434138,145692400018,2118364506746,31529605958892,

%U 480186833802260,7483472464151002,119397596900634238,1951747376021480874,32721008993895160926,563260078608381337148,9967709437187109520736,181544883799333028286098,3406426523158387599683478,65894531117591548919270114

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

%C Compare to: G(x) = Sum_{n>=0} (1 + x*G(x)^k)^n / 2^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.

%H Paul D. Hanna, <a href="/A300050/b300050.txt">Table of n, a(n) for n = 0..100</a>

%F G.f. A(x) satisfies:

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

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

%e G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 250*x^4 + 2558*x^5 + 28594*x^6 + 340486*x^7 + 4255344*x^8 + 55300776*x^9 + 742732646*x^10 + ...

%e such that

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

%e Also, due to a series identity,

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

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

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A302103, A302104, A302105, A300279.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 27 2018