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%I #9 May 08 2019 19:34:52
%S 1,7,29,199,1799,17227,186329,2206267,28014806,379929320,5463954442,
%T 82846605311,1319217639773,21986394800481,382391651777315,
%U 6923247705041891,130203668578601251,2538723618787364104,51232176790185969490,1068400287907868926763,22992012848929364898826,509930616374506270683390,11641797557959754404475921,273291680187022711826902024
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^(4*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)^(4*n) - A(x))^(n+1).
%F (2) 1 + x = Sum_{n>=0} x^n * (1+x)^(4*n*(n-1)) / (1 + x*(1+x)^(4*n)*A(x))^(n+1).
%F (3) 1 = Sum_{n>=0} x^n * (1-x)^(8*n+2) / ((1-x)^(4*n+1) - x*A(x/(1-x)))^(n+1).
%F (4) 1 = Sum_{n>=0} x^n * (1 - (1-x)^(4*n-4) * A(x/(1-x)))^n / (1-x)^(4*n^2-3*n-1)).
%e G.f.: A(x) = x + 7*x^2 + 29*x^3 + 199*x^4 + 1799*x^5 + 17227*x^6 + 186329*x^7 + 2206267*x^8 + 28014806*x^9 + 379929320*x^10 + 5463954442*x^11 + ...
%e such that
%e 1 = (1 - A(x)) + x*((1+x)^4 - A(x))^2 + x^2*((1+x)^8 - A(x))^3 + x^3*((1+x)^12 - A(x))^4 + x^4*((1+x)^16 - A(x))^5 + x^5*((1+x)^20 - A(x))^6 + x^6*((1+x)^24 - A(x))^7 + x^7*((1+x)^28 - A(x))^8 + ...
%e also
%e 1 + x = 1/(1 + x*A(x)) + x/(1 + x*(1+x)^4*A(x))^2 + x^2*(1+x)^8/(1 + x*(1+x)^8*A(x))^3 + x^3*(1+x)^24/(1 + x*(1+x)^12*A(x))^4 + x^4*(1+x)^48/(1 + x*(1+x)^16*A(x))^5 + x^5*(1+x)^80/(1 + x*(1+x)^20*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))^(4*m) - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A307940, A307952, A307953, A307955.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 07 2019
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