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%I #12 May 30 2018 12:34:17
%S 1,1,-1,-4,-2,18,40,-30,-289,-346,1151,4319,658,-27009,-55352,76422,
%T 521723,474100,-2701432,-8516598,2522919,65164743,108748950,
%U -254553384,-1279831072,-665253375,7987959534,20547223364,-18007631653,-186391127742,-235366676097,912066105170,3550265215708,259991506989,-25935784814796,-53475930511840,87779402689817,575950379415007,488006663591490
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (x^n - A(x))^n.
%H Paul D. Hanna, <a href="/A305135/b305135.txt">Table of n, a(n) for n = 1..1030</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} (x^n - A(x))^n.
%F (2) 1 = Sum_{n>=0} x^(n^2) / (1 + x^n*A(x))^(n+1).
%e G.f.: A(x) = x + x^2 - x^3 - 4*x^4 - 2*x^5 + 18*x^6 + 40*x^7 - 30*x^8 - 289*x^9 - 346*x^10 + 1151*x^11 + 4319*x^12 + 658*x^13 - 27009*x^14 - 55352*x^15 + 76422*x^16 + 521723*x^17 + 474100*x^18 - 2701432*x^19 - 8516598*x^20 + 2522919*x^21 + 65164743*x^22 + 108748950*x^23 - 254553384*x^24 - 1279831072*x^25 + ...
%e such that
%e 1 = 1 + (x - A(x)) + (x^2 - A(x))^2 + (x^3 - A(x))^3 + (x^4 - A(x))^4 + (x^5 - A(x))^5 + (x^6 - A(x))^6 + (x^7 - A(x))^7 + ... + (x^n - A(x))^n + ...
%e Also,
%e 1 = 1/(1 + A(x)) + x/(1 + x*A(x))^2 + x^4/(1 + x^2*A(x))^3 + x^9/(1 + x^3*A(x))^4 + x^16/(1 + x^4*A(x))^5 + x^25/(1 + x^5*A(x))^6 + x^36/(1 + x^6*A(x))^7 + x^49/(1 + x^7*A(x))^8 + x^64/(1 + x^8*A(x))^9 + ... + x^(n^2)/(1 + x^n*A(x))^(n+1) + ...
%e RELATED SERIES.
%e Series_Reversion(A(x)) = x - x^2 + 3*x^3 - 6*x^4 + 16*x^5 - 46*x^6 + 142*x^7 - 471*x^8 + 1606*x^9 - 5616*x^10 + 19946*x^11 + ... + A305136(n)*x^n + ...
%e SPECIFIC VALUES.
%e (1) Although the series diverges at x = 1/2, it can be evaluated formally.
%e Let b = A(1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
%e then 1 = Sum_{n>=0} (1/2^n - b)^n
%e and 1 = Sum_{n>=0} 2^n/(2^n + b)^(n+1).
%e (2) The series converges at x = 1/3 ;
%e let t = A(1/3) = 0.3773536554501605670840638296726444190565890261333229000698630...
%e then 1 = Sum_{n>=0} (1/3^n - t)^n
%e and 1 = Sum_{n>=0} 3^n/(3^n + t)^(n+1).
%o (PARI) /* From 1 = Sum_{n>=0} (x^n - A(x))^n */
%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A+1, (x^m - x*Ser(A))^m ) )[#A+1] ); A[n]}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) /* From 1 = Sum_{n>=0} x^(n^2) / (1 + x^n * A(x))^(n+1) */
%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0,sqrtint(#A+1), x^(m^2)/(1 + x^m*x*Ser(A))^(m+1) ) )[#A+1] ); A[n]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A305136.
%K sign
%O 1,4
%A _Paul D. Hanna_, May 30 2018
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