This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357223 #5 Sep 19 2022 11:16:10
%S A357223 1,1,4,25,164,1177,8887,69748,563232,4649672,39063521,332904462,
%T A357223 2870862974,25005954906,219675658337,1944131038267,17316793719372,
%U A357223 155122164103293,1396584226654493,12630315100857638,114687815080027358,1045218902425525155,9557367319452886097
%N A357223 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357223 G.f. A(x) satisfies:
%F A357223 (1) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
%F A357223 (2) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
%F A357223 (3) x*A(x)^3 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
%F A357223 (4) -x*A(x)^4 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
%e A357223 G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 164*x^4 + 1177*x^5 + 8887*x^6 + 69748*x^7 + 563232*x^8 + 4649672*x^9 + 39063521*x^10 + ...
%e A357223 such that
%e A357223 x*A(x)^3 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...
%o A357223 (PARI) {a(n,p=3) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A357223 A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
%o A357223 for(n=0, 30, print1(a(n), ", "))
%Y A357223 Cf. A355357, A357221, A357222, A357224, A357225, A357226.
%K A357223 nonn
%O A357223 0,3
%A A357223 _Paul D. Hanna_, Sep 18 2022