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%I #9 Sep 20 2022 21:22:32
%S 1,1,4,25,162,1160,8731,68364,550707,4535402,38012170,323168946,
%T 2780229079,24158457026,211721412339,1869239684558,16609750957942,
%U 148431230687412,1333134683364035,12027524448579488,108951760865234373,990555733683233240,9035754580314840475
%N Coefficients in the power series A(x) such that: A(x)^2 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%C Compare to A357152.
%C Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
%C Related identity: 0 = Sum_{n=-oo..+oo} x^(k*n) * (y - x^(n+1-k))^n, which holds for any positive integer k and real y.
%H Paul D. Hanna, <a href="/A357162/b357162.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F (1) A(x)^2 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%F (2) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
%F (3) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
%F (4) -A(x)^5 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
%F (5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*A(x))^(n+1) / A(x)^n.
%F (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+2))^n.
%e G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 162*x^4 + 1160*x^5 + 8731*x^6 + 68364*x^7 + 550707*x^8 + 4535402*x^9 + 38012170*x^10 + ...
%e such that
%e A(x)^2 = ... + x^(-4)*(1 - 1/x^3)^(-1)/A(x)^2 + x^(-1)/A(x) + x^2*(1 - 1/x) + x^5*0*A(x) + x^8*(1 - x)^3*A(x)^2 + x^11*(1 - x^2)^4*A(x)^3 + ... + x^(3*n+2)*(1 - x^(n-1))^(n+1)*A(x)^n + ...
%e also
%e -A(x)^5 = ... + x^(-4)*(A(x) - 1/x^3)^(-1)*A(x)^2 + x^(-1)*A(x) + x^2*(A(x) - 1/x) + x^5*(A(x) - 1)^2/A(x) + x^8*(A(x) - x)^3/A(x)^2 + x^11*(A(x) - x^2)^4/A(x)^3 + ... + x^(3*n+2)*(A(x) - x^(n-1))^(n+1)/A(x)^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
%o A[#A] = polcoeff(Ser(A)^2 - sum(n=-#A\3-2,#A\3+2, x^(3*n+2) * (1 - x^(n-1) +x*O(x^#A))^(n+1) * Ser(A)^n ),#A-2); );A[n+1]}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A357152, A357160, A357161, A357163, A357164, A357165.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 17 2022
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