cp's OEIS Frontend

A357165 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.

%I A357165 #9 Sep 20 2022 21:50:34
%S A357165 1,1,7,73,859,11083,151369,2151961,31510682,471993401,7198166363,
%T A357165 111390268227,1744706996712,27606853938808,440638645554932,
%U A357165 7086053148425023,114700710907449375,1867353232898846998,30556409451787334011,502291724376632138667,8290605658533141188978
%N A357165 Coefficients in the power series A(x) such that: A(x)^5 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%C A357165 Compare to A357155.
%C A357165 Related identity: 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1).
%C A357165 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 A357165 Paul D. Hanna, <a href="/A357165/b357165.txt">Table of n, a(n) for n = 0..300</a>
%F A357165 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357165 (1) A(x)^5 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1))^(n+1) * A(x)^n.
%F A357165 (2) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+2))^n * A(x)^n ).
%F A357165 (3) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+2)*A(x))^n.
%F A357165 (4) -A(x)^8 = Sum_{n=-oo..+oo} x^(3*n+2) * (A(x) - x^(n-1))^(n+1) / A(x)^n.
%F A357165 (5) 0 = Sum_{n=-oo..+oo} x^(3*n+2) * (1 - x^(n-1)*A(x))^(n+1) / A(x)^n.
%F A357165 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+2))^n.
%e A357165 G.f.: A(x) = 1 + x + 7*x^2 + 73*x^3 + 859*x^4 + 11083*x^5 + 151369*x^6 + 2151961*x^7 + 31510682*x^8 + 471993401*x^9 + 7198166363*x^10 + ...
%e A357165 such that
%e A357165 A(x)^5 = ... + 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 A357165 also
%e A357165 -A(x)^8 = ... + 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 A357165 (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
%o A357165 A[#A] = polcoeff(Ser(A)^5 - 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 A357165 for(n=0,30, print1(a(n),", "))
%Y A357165 Cf. A357155, A357160, A357161, A357162, A357163, A357164.
%K A357165 nonn
%O A357165 0,3
%A A357165 _Paul D. Hanna_, Sep 17 2022