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G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(2*n) + (-1)^n*A(x))^n.
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%I #5 Apr 07 2022 12:11:13

%S 1,2,3,5,10,21,45,100,225,515,1190,2780,6550,15555,37187,89428,216163,

%T 524890,1279727,3131526,7688383,18933352,46754406,115750501,287238398,

%U 714340089,1780091533,4444198347,11114847215,27843451884,69856594565,175515654498

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

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

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

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

%e G.f.: A(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 10*x^5 + 21*x^6 + 45*x^7 + 100*x^8 + 225*x^9 + 515*x^10 + 1190*x^11 + 2780*x^12 + ...

%e where

%e 1 - x = 1 + (x^2 - A(x)) + (x^4 + A(x))^2 + (x^6 - A(x))^3 + (x^8 + A(x))^4 + (x^10 - A(x))^5 + (x^12 + A(x))^6 + ...

%e Also,

%e 1 - x = 1/(1 + A(x)) + x^2/(1 - x^2*A(x))^2 + x^8/(1 + x^4*A(x))^3 + x^18/(1 - x^6*A(x))^4 + x^32/(1 + x^8*A(x))^5 + ...

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

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

%o for(n=0,40,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\2), x^(2*m^2)/(1 + (-x^2)^m*x*Ser(A))^(m+1) ),#A));A[n+1]}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A317997, A352817, A352819, A352820, A352821.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Apr 05 2022