%I #12 Apr 11 2018 18:55:47
%S 1,1,5,47,641,11283,243755,6236425,184344339,6180934293,231761841775,
%T 9609095960569,436486983640191,21556547150620421,1149991421265821805,
%U 65903887072762826847,4037804462230246970067,263376035279468797850997,18222095466457124888031163,1332861882984996470788507485,102768790354267787018489100069,8330655428164879820409112566087
%N G.f.: Sum_{n>=0} (1+x)^n * ((1+x)^n + (1+2*x)^n)^n / (2*(1+x)^n + (1+2*x)^n)^(n+1).
%H Paul D. Hanna, <a href="/A302616/b302616.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n equals:
%F (1) Sum_{n>=0} (1+x)^n * ((1+2*x)^n + (1+x)^n)^n / (2*(1+x)^n + (1+2*x)^n)^(n+1).
%F (2) Sum_{n>=0} (1+x)^n * ((1+2*x)^n - (1+x)^n)^n / (2*(1+x)^n - (1+2*x)^n)^(n+1).
%e G.f.: A(x) = 1 + x + 5*x^2 + 47*x^3 + 641*x^4 + 11283*x^5 + 243755*x^6 + 6236425*x^7 + 184344339*x^8 + 6180934293*x^9 + ...
%e such that
%e A(x) = 1/3 + (1+x)*((1+x) + (1+2*x))/(2*(1+x) + (1+2*x))^2 + (1+x)^2*((1+x)^2 + (1+2*x)^2)^2/(2*(1+x)^2 + (1+2*x)^2)^3 + (1+x)^3*((1+x)^3 + (1+2*x)^3)^3/(2*(1+x)^3 + (1+2*x)^3)^4 + (1+x)^4*((1+x)^4 + (1+2*x)^4)^4/(2*(1+x)^4 + (1+2*x)^4)^5 + ...
%e Also,
%e A(x) = 1 + (1+x)*((1+2*x) - (1+x))/(2*(1+x) - (1+2*x))^2 + (1+x)^2*((1+2*x)^2 - (1+x)^2)^2/(2*(1+x)^2 - (1+2*x)^2)^3 + (1+x)^3*((1+2*x)^3 - (1+x)^3)^3/(2*(1+x)^3 - (1+2*x)^3)^4 + (1+x)^4*((1+2*x)^4 - (1+x)^4)^4/(2*(1+x)^4 - (1+2*x)^4)^5 + ...
%o (PARI) {a(n) = my(A=1,o=x*O(x^n)); A = sum(m=0,n, (1+x)^m*((1+2*x)^m - (1+x)^m +o)^m/(2*(1+x)^m - (1+2*x)^m +o)^(m+1)); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 11 2018