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A357405 Coefficients in the power series A(x) such that: 5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%I A357405 #6 Sep 27 2022 12:49:10
%S A357405 1,5,50,630,8825,132490,2084115,33903705,565697930,9627904690,
%T A357405 166493454330,2917050253615,51670197054515,923774673549045,
%U A357405 16647699155752645,302098954307654995,5515438344643031325,101237254225602624790,1867129260849076888865,34583287418814030368150
%N A357405 Coefficients in the power series A(x) such that: 5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%C A357405 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%C A357405 a(n) = Sum_{k=0..n} A357400(n,k) * 5^k, for n >= 0.
%H A357405 Paul D. Hanna, <a href="/A357405/b357405.txt">Table of n, a(n) for n = 0..300</a>
%F A357405 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357405 (1) 5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%F A357405 (2) 5*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
%F A357405 (3) -5*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
%F A357405 (4) -5*A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
%F A357405 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
%F A357405 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
%e A357405 G.f.: A(x) = 1 + 5*x + 50*x^2 + 630*x^3 + 8825*x^4 + 132490*x^5 + 2084115*x^6 + 33903705*x^7 + 565697930*x^8 + 9627904690*x^9 + 166493454330*x^10 + ...
%e A357405 such that
%e A357405 5 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
%e A357405 also
%e A357405 -5*A(x)^3 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...
%o A357405 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357405 A[#A] = polcoeff(5 - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); A[n+1]}
%o A357405 for(n=0, 30, print1(a(n), ", "))
%Y A357405 Cf. A357400, A356783, A357402, A357403, A357404.
%K A357405 nonn
%O A357405 0,2
%A A357405 _Paul D. Hanna_, Sep 26 2022