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 A354663 #9 Jul 22 2022 17:44:19
%S A354663 2,9,108,1848,36306,771768,17280096,401451192,9587095686,233892105912,
%T A354663 5804193409056,146051807458320,3717875447707254,95571022734750600,
%U A354663 2477365983601721280,64684289495622383472,1699638032224106092368,44909438746576707103608
%N A354663 G.f. A(x) satisfies: 3 = Sum_{n=-oo..oo} (-x)^(n*(n+1)/2) * A(x)^(n*(n-1)/2).
%H A354663 Paul D. Hanna, <a href="/A354663/b354663.txt">Table of n, a(n) for n = 0..400</a>
%F A354663 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A354663 (1) 3 = Sum_{n=-oo..oo} (-x)^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
%F A354663 (2) 3 = Sum_{n>=0} (-x)^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
%F A354663 (3) 3 = Sum_{n>=0} (-1)^(n*(n+1)/2) * A(x)^(n*(n-1)/2) * (1 + A(x)^(2*n+1)) * x^(n*(n+1)/2).
%F A354663 (4) 3 = Product_{n>=1} (1 - (-x)^n*A(x)^n) * (1 + (-x)^(n-1)*A(x)^n) * (1 + (-x)^n*A(x)^(n-1)), by the Jacobi triple product identity.
%F A354663 a(n) = (-1)^n * Sum_{k=0..2*n+1} A354649(n,k)*3^k, for n >= 0.
%F A354663 a(n) = -Sum_{k=0..2*n+1} A354650(n,k)*(-3)^k, for n >= 0.
%e A354663 G.f.: A(x) = 2 + 9*x + 108*x^2 + 1848*x^3 + 36306*x^4 + 771768*x^5 + 17280096*x^6 + 401451192*x^7 + 9587095686*x^8 + 233892105912*x^9 + ...
%e A354663 such that A = A(x) satisfies:
%e A354663 (1) 3 = ... + x^36*A^28 + x^28*A^21 - x^21*A^15 - x^15*A^10 + x^10*A^6 + x^6*A^3 - x^3*A - x + 1 + A - x*A^3 - x^3*A^6 + x^6*A^10 + x^10*A^15 - x^15*A^21 - x^21*A^28 + x^28*A^36 +--+ ...
%e A354663 (2) 3 = (1-x) + (1-x^3)*A - x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 + x^10*(1-x^11)*A^15 - x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
%e A354663 (3) 3 = (1+A) - (1+A^3)*x - A*(1+A^5)*x^3 + A^3*(1+A^7)*x^6 + A^6*(1+A^9)*x^10 - A^10*(1+A^11)*x^15 - A^15*(1+A^13)*x^21 + A^21*(1+A^15)*x^28 + ...
%e A354663 (4) 3 = (1 + x*A)*(1 + A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 + x^2*A) * (1 + x^3*A^3)*(1 + x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 + x^4*A^3) * (1 + x^5*A^5)*(1 + x^4*A^5)*(1 - x^5*A^4) * ...
%o A354663 (PARI) {a(n) = my(A=[2]); for(i=1,n, A = concat(A,0);
%o A354663 A[#A] = -polcoeff(-3 + sum(m=0,sqrtint(2*#A+9), (-x)^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );H=A;A[n+1]}
%o A354663 for(n=0,30,print1(a(n),", "))
%Y A354663 Cf. A354649, A354650, A354661, A354662, A354664, A268299, A354652, A354653, A354654.
%K A354663 nonn
%O A354663 0,1
%A A354663 _Paul D. Hanna_, Jun 02 2022