cp's OEIS Frontend

A357155 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%I A357155 #9 Sep 16 2022 22:10:56
%S A357155 1,1,7,71,832,10660,144684,2043814,29736131,442562703,6706068107,
%T A357155 103109044005,1604621459651,25226987525340,400062373648799,
%U A357155 6392118111706099,102801779216363982,1662854341556813731,27034758217304814579,441537893821034707720,7240848432876171585800
%N A357155 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%C A357155 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%H A357155 Paul D. Hanna, <a href="/A357155/b357155.txt">Table of n, a(n) for n = 0..400</a>
%F A357155 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357155 (1) A(x)^5 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%F A357155 (2) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
%F A357155 (3) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
%F A357155 (4) -A(x)^8 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
%F A357155 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
%F A357155 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
%e A357155 G.f.: A(x) = 1 + x + 7*x^2 + 71*x^3 + 832*x^4 + 10660*x^5 + 144684*x^6 + 2043814*x^7 + 29736131*x^8 + 442562703*x^9 + 6706068107*x^10 + 103109044005*x^11 + 1604621459651*x^12 + ...
%e A357155 such that
%e A357155 A(x)^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 A357155 also
%e A357155 -A(x)^8 = ... + 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 A357155 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357155 A[#A] = polcoeff(Ser(A)^5 - sum(n=-#A\2-1, #A\2+1, x^(2*n+1) * (1 - x^n +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
%o A357155 for(n=0, 30, print1(a(n), ", "))
%Y A357155 Cf. A356783, A357151, A357152, A357153, A357154.
%K A357155 nonn
%O A357155 0,3
%A A357155 _Paul D. Hanna_, Sep 16 2022