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%I #12 May 08 2019 19:31:32
%S 1,3,7,39,218,1396,10078,78369,655415,5833338,54863836,542721779,
%T 5623476082,60831556079,685114308524,8014714349561,97189873705285,
%U 1219512416998790,15808537423941847,211404384080948562,2912638538651962032,41294363532894786740,601795320633550518240,9005722875874046697058,138257703952334108620495,2175595836996606841813378,35061246912965660203462227
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^(2*n) - A(x))^(n+1), where A(0) = 0.
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} x^n * ((1+x)^(2*n) - A(x))^(n+1).
%F (2) 1 + x = Sum_{n>=0} x^n * (1+x)^(2*n*(n-1)) / (1 + x*(1+x)^(2*n)*A(x))^(n+1).
%F (3) 1 = Sum_{n>=0} x^n * (1-x)^(4*n+2) / ((1-x)^(2*n+1) - x*A(x/(1-x)))^(n+1).
%F (4) 1 = Sum_{n>=0} x^n * (1 - (1-x)^(2*n-2) * A(x/(1-x)))^n / (1-x)^(2*n^2-n-1)).
%e G.f.: A(x) = x + 3*x^2 + 7*x^3 + 39*x^4 + 218*x^5 + 1396*x^6 + 10078*x^7 + 78369*x^8 + 655415*x^9 + 5833338*x^10 + 54863836*x^11 + 542721779*x^12 + ...
%e such that
%e 1 = (1 - A(x)) + x*((1+x)^2 - A(x))^2 + x^2*((1+x)^4 - A(x))^3 + x^3*((1+x)^6 - A(x))^4 + x^4*((1+x)^8 - A(x))^5 + x^5*((1+x)^10 - A(x))^6 + x^6*((1+x)^12 - A(x))^7 + x^7*((1+x)^14 - A(x))^8 + ...
%e also
%e 1 + x = 1/(1 + x*A(x)) + x/(1 + x*(1+x)^2*A(x))^2 + x^2*(1+x)^4/(1 + x*(1+x)^4*A(x))^3 + x^3*(1+x)^12/(1 + x*(1+x)^6*A(x))^4 + x^4*(1+x)^24/(1 + x*(1+x)^8*A(x))^5 + x^5*(1+x)^40/(1 + x*(1+x)^10*A(x))^6 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1+x +x*O(x^#A))^(2*m) - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A307940, A307953, A307954, A307955.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 07 2019
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