%I #10 Feb 16 2021 12:42:20
%S 1,2,4,8,17,38,90,225,593,1642,4762,14418,45443,148713,504149,1766929,
%T 6390719,23815167,91306967,359694456,1454213025,6027213530,
%U 25583995336,111118605582,493407322279,2238131105769,10363617299669,48954143141360
%N G.f.: Sum_{n>=0} x^n/(1 - x*(1+x)^(n+1)).
%C The g.f. of this sequence is motivated by the following identity:
%C Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
%C here, p = x, q = x*(1+x), and r = (1+x).
%F G.f.: Sum_{n>=0} x^n / (1 - x*(1+x)^(n+1)).
%F G.f.: Sum_{n>=0} x^n*(1+x)^n / (1 - x*(1+x)^n).
%F G.f.: Sum_{n>=0} x^(2*n) * (1+x)^(n^2+n) * (1 - x^2*(1+x)^(2*n+1)) / ((1 - x*(1+x)^(n))*(1 - x*(1+x)^(n+1))).
%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 17*x^4 + 38*x^5 + 90*x^6 + 225*x^7 + 593*x^8 + 1642*x^9 + 4762*x^10 + 14418*x^11 + 45443*x^12 + ...
%e where
%e A(x) = 1/(1 - x*(1+x)) + x/(1 - x*(1+x)^2) + x^2/(1 - x*(1+x)^3) + x^3/(1 - x*(1+x)^4) + x^4/(1 - x*(1+x)^5) + x^5/(1 - x*(1+x)^6) + ...
%e also
%e A(x) = 1/(1 - x) + x*(1+x)/(1 - x*(1+x)) + x^2*(1+x)^2/(1 - x*(1+x)^2) + x^3*(1+x)^3/(1 - x*(1+x)^3) + x^4*(1+x)^4/(1 - x*(1+x)^4) + ...
%o (PARI) {a(n) = my(A = sum(m=0, n, x^m /(1 - x*(1+x)^(m+1) +x*O(x^n)) )); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) {a(n) = my(A = sum(m=0, n, x^m*(1+x)^m /(1 - x*(1+x)^m +x*O(x^n)) )); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A340775.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 20 2021