cp's OEIS Frontend

A357203 Coefficients in the power series A(x) such that: A(x)^3 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.

%I A357203 #7 Sep 20 2022 00:01:59
%S A357203 1,1,3,18,111,800,5990,46995,379090,3129713,26301576,224282112,
%T A357203 1935668344,16876028036,148410725830,1314933853171,11726585616205,
%U A357203 105178923513494,948185788906100,8586757756571261,78079244607685021,712592590813142079,6525273550226573555
%N A357203 Coefficients in the power series A(x) such that: A(x)^3 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
%C A357203 Compare to A357153 and A357163.
%C A357203 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1).
%C A357203 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 A357203 Paul D. Hanna, <a href="/A357203/b357203.txt">Table of n, a(n) for n = 0..300</a>
%F A357203 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A357203 (1) A(x)^3 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1))^(n+1) * A(x)^n.
%F A357203 (2) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^n)^n * A(x)^n ).
%F A357203 (3) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^n*A(x))^n.
%F A357203 (4) -A(x)^6 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(n+1))^(n+1) / A(x)^n.
%F A357203 (5) 0 = Sum_{n=-oo..+oo} x^n * (1 - x^(n+1)*A(x))^(n+1) / A(x)^n.
%F A357203 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^n)^n.
%e A357203 G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 111*x^4 + 800*x^5 + 5990*x^6 + 46995*x^7 + 379090*x^8 + 3129713*x^9 + 26301576*x^10 + ...
%e A357203 such that
%e A357203 A(x)^3 = ... + x^(-2)*(1 - 1/x)^(-1)/A(x)^2 + x^(-1)/A(x) + (1 - x) + x*(1 - x^2)*A(x) + x^2*(1 - x^3)^3*A(x)^2 + x^3*(1 - x^4)^4*A(x)^3 + ... + x^n*(1 - x^(n+1))^(n+1)*A(x)^n + ...
%e A357203 also
%e A357203 -A(x)^6 = ... + x^(-2)*(A(x) - 1/x)^(-1)*A(x)^2 + x^(-1)*A(x) + (A(x) - x) + x*(A(x) - x^2)^2/A(x) + x^2*(A(x) - x^3)^3/A(x)^2 + x^3*(A(x) - x^4)^4/A(x)^3 + ... + x^n*(A(x) - x^(n+1))^(n+1)/A(x)^n + ...
%o A357203 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357203 A[#A] = polcoeff(Ser(A)^3 - sum(n=-#A-2, #A+2, x^(n) * (1 - x^(n+1) +x*O(x^#A))^(n+1) * Ser(A)^n  ), #A-2); ); A[n+1]}
%o A357203 for(n=0, 30, print1(a(n), ", "))
%Y A357203 Cf. A357153, A357163, A357200, A357201, A357202, A357204, A357205.
%K A357203 nonn
%O A357203 0,3
%A A357203 _Paul D. Hanna_, Sep 17 2022