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G.f. A(x) satisfies: 1 = Sum_{n>=0} (A(x)^n - x)^n.
2

%I #6 May 30 2018 01:56:39

%S 1,-1,3,-6,16,-46,142,-471,1606,-5616,19946,-71659,260044,-951532,

%T 3507916,-13018488,48601126,-182406035,687851096,-2604964819,

%U 9903253875,-37780174088,144584919872,-554927186056,2135497844772,-8237988842334,31850890373750,-123403823544802,479047178570539,-1863001134095713,7257404420481641,-28316271490365464,110645779544592100

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (A(x)^n - x)^n.

%H Paul D. Hanna, <a href="/A305136/b305136.txt">Table of n, a(n) for n = 1..1030</a>

%F G.f. A(x) satisfies:

%F (1) 1 = Sum_{n>=0} (A(x)^n - x)^n.

%F (2) 1 = Sum_{n>=0} A(x)^(n^2) / (1 + x*A(x)^n)^(n+1).

%e G.f.: 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 - 71659*x^12 + 260044*x^13 - 951532*x^14 + 3507916*x^15 - 13018488*x^16 + 48601126*x^17 - 182406035*x^18 + ...

%e such that

%e 1 = 1 + (A(x) - x) + (A(x)^2 - x)^2 + (A(x)^3 - x)^3 + (A(x)^4 - x)^4 + (A(x)^5 - x)^5 + (A(x)^6 - x)^6 + (A(x)^7 - x)^7 + ... + (A(x)^n - x)^n + ...

%e Also,

%e 1 = 1/(1 + x) + A(x)/(1 + x*A(x))^2 + A(x)^4/(1 + x*A(x)^2)^3 + A(x)^9/(1 + x*A(x)^3)^4 + A(x)^16/(1 + x*A(x)^4)^5 + A(x)^25/(1 + x*A(x)^5)^6 + A(x)^36/(1 + x*A(x)^6)^7 + A(x)^49/(1 + x*A(x)^7)^8 + ... + A(x)^(n^2)/(1 + x*A(x)^n)^(n+1) + ...

%e RELATED SERIES.

%e Series_Reversion(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 + ... + A305135(n)*x^n + ...

%o (PARI) {a(n) = my(A,V=[1]); for(i=1, n, V=concat(V, 0); V[#V] = Vec( sum(m=0, #V+1, (x^m - x*Ser(V))^m ) )[#V+1] ); A = serreverse(x*Ser(V)); polcoeff(A,n)}

%o for(n=1, 30, print1(a(n), ", "))

%o (PARI) {a(n) = my(A,V=[1]); for(i=1, n, V=concat(V, 0); V[#V] = Vec( sum(m=0, sqrtint(#V+1), x^(m^2)/(1 + x^m*x*Ser(V))^(m+1) ) )[#V+1] ); A = serreverse(x*Ser(V)); polcoeff(A,n)}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A305135.

%K sign

%O 1,3

%A _Paul D. Hanna_, May 30 2018