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G.f. A(x) satisfies: 1 = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * ((1+x)^n + A(x))^(n+1).
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%I #8 Feb 10 2021 11:12:08

%S 1,4,11,29,76,207,592,1780,5617,18365,61465,209173,720636,2505401,

%T 8773807,30919540,109581804,390360666,1397013258,5020578459,

%U 18111752521,65564995435,238097178090,867136619465,3166405246169,11590402816879,42520849230446

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

%e G.f.: A(x) = x + 4*x^2 + 11*x^3 + 29*x^4 + 76*x^5 + 207*x^6 + 592*x^7 + 1780*x^8 + 5617*x^9 + 18365*x^10 + 61465*x^11 + 209173*x^12 + ...

%e where

%e 1 = (1 + A(x)) - x*((1+x) + A(x))^2 + x^3*((1+x)^2 + A(x))^3 - x^6*((1+x)^3 + A(x))^4 + x^10*((1+x)^4 + A(x))^5 - x^15*((1+x)^5 + A(x))^6 + ...

%e SPECIFIC VALUES.

%e At x = -1/2, 1 = Sum_{n>=0} (-1)^n * (-1/2)^(n*(n+1)/2) * (1/2^n + a)^(n+1), where a = 0.0849569238946083173319941875462909358145...

%e At x = 1/2, 1 = Sum_{n>=0} (-1)^n * (1/2)^(n*(n+1)/2) * ((3/2)^n + a)^(n+1), where a = 12.925265220825824331002181448511410832160...

%o (PARI) {a(n) = A=[1]; for(i=1, n, A = concat(A,0);

%o A[#A] = -polcoeff( sum(m=0,sqrtint(2*#A)+1, (-1)^m * x^(m*(m+1)/2)*((1+x +x*O(x^#A))^m + x*Ser(A))^(m+1) ),#A) );A[n]}

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

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 09 2021