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

%I #4 Jul 24 2022 01:45:48

%S 1,2,2,5,13,34,100,310,973,3124,10251,34086,114610,389205,1332788,

%T 4596587,15952281,55667566,195209636,687535017,2431044066,8626448629,

%U 30709429207,109645265403,392536229012,1408793986056,5067691347863,18268127956618,65983199912542

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

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

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

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

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 5*x^3 + 13*x^4 + 34*x^5 + 100*x^6 + 310*x^7 + 973*x^8 + 3124*x^9 + 10251*x^10 + 34086*x^11 + 114610*x^12 + ...

%e where

%e 0 = ... + x^9/(1/x^3 - A(x))^2 + x^4/(1/x^2 - A(x)) + x + (1 - A(x)) + x*(x - A(x))^2 + x^4*(x^2 - A(x))^3 + x^9*(x^3 - A(x))^4 + x^16*(x^4 - A(x))^5 + ... + x^(n^2)*(x^n - A(x))^(n+1) + ...

%o (PARI) {a(n) = my(A=[1],M); for(i=1,n, A=concat(A,0); M = ceil(sqrt(2*(#A)+9));

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

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 22 2022