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 A363184 #11 May 23 2023 14:28:45
%S A363184 1,4,18,88,452,2388,12872,70520,391630,2199816,12476024,71341184,
%T A363184 410864744,2381026908,13874518912,81244555896,477825991140,
%U A363184 2821333839872,16718050009866,99385412418648,592575029005992,3542752436877800,21233468105000280,127555885796445432
%N A363184 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^(n+1).
%C A363184 Conjecture: g.f. A(x) == theta_3(x^2) (mod 4); a(n) == 2 (mod 4) if n = 2*k^2 for integer k > 0, and a(n) == 0 (mod 4) if floor(n/2) is nonsquare.
%H A363184 Paul D. Hanna, <a href="/A363184/b363184.txt">Table of n, a(n) for n = 0..300</a>
%F A363184 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A363184 (1) 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^(n+1).
%F A363184 (2) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 4*A(x)*x^(2*n+1))^(n-1).
%F A363184 (3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^n.
%F A363184 (4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (4*A(x) + x^(2*n-1))^(n-1).
%F A363184 (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 4*A(x)*x^(2*n+1))^n.
%F A363184 a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 4^(n-2*k) for n >= 0.
%e A363184 G.f.: A(x) = 1 + 4*x + 18*x^2 + 88*x^3 + 452*x^4 + 2388*x^5 + 12872*x^6 + 70520*x^7 + 391630*x^8 + 2199816*x^9 + 12476024*x^10 + ...
%o A363184 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A363184 A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (4*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/4); A[n+1]}
%o A363184 for(n=0, 30, print1(a(n), ", "))
%Y A363184 Cf. A363142, A363182, A363183, A363185.
%Y A363184 Cf. A359670.
%K A363184 nonn
%O A363184 0,2
%A A363184 _Paul D. Hanna_, May 20 2023