%I #10 Apr 06 2022 18:59:35
%S 1,4,15,62,263,1153,5187,23792,110898,523773,2501268,12057407,
%T 58593831,286743949,1411905287,6989973590,34773216944,173737947911,
%U 871442154413,4386482848975,22150822685669,112185906664804,569713055956736,2900350345874632,14799219791196091
%N G.f. A(x) satisfies: 1 + x = Sum_{n>=0} (-1)^n * (x^n + A(x))^(n+1).
%H Paul D. Hanna, <a href="/A352815/b352815.txt">Table of n, a(n) for n = 1..500</a>
%F G.f. A(x) satisfies:
%F (1) 1 + x = Sum_{n>=0} (-1)^n * (x^n + A(x))^(n+1).
%F (2) -x = Sum_{n>=0} (-1)^n * x^(n*(n-1)) / (1 + x^n*A(x))^(n+1).
%e G.f.: A(x) = x + 4*x^2 + 15*x^3 + 62*x^4 + 263*x^5 + 1153*x^6 + 5187*x^7 + 23792*x^8 + 110898*x^9 + 523773*x^10 + ...
%e where
%e 1 + x = (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 + ...
%e also
%e -x = 1/(1 + A(x)) - 1/(1 + x*A(x))^2 + x^2/(1 + x^2*A(x))^3 - x^6/(1 + x^3*A(x))^4 + x^12/(1 + x^4*A(x))^5 - x^20/(1 + x^5*A(x))^6 + ...
%e Specific values.
%e A(x) = 1 at x = 0.1834970136530040531685106821803389905413247357336272...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = -polcoeff( sum(m=0,#A, (-1)^m*(x^m + x*Ser(A))^(m+1) ),#A));A[n]}
%o for(n=1,40,print1(a(n),", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,sqrtint(#A)+1, (-1)^m*x^(m*(m-1))/(1 + x^m*x*Ser(A))^(m+1) ),#A));A[n]}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A317997.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Apr 04 2022