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A379204 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).

%I A379204 #7 Dec 20 2024 23:38:56
%S A379204 1,6,52,572,7154,96444,1365480,20015404,301104656,4622137698,
%T A379204 72110068424,1140008607808,18223311950352,294049155429240,
%U A379204 4783093039542544,78348659072215696,1291254702576739650,21396346604365855060,356250789435149406252,5957201829333106382128,100003077199160840926640
%N A379204 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).
%C A379204 Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
%C A379204 Conjecture: a(n) is even for n > 1.
%C A379204 Conjecture: a(n) == 2 (mod 4) iff n = (k-1)^2 + 1 for some k > 1.
%H A379204 Paul D. Hanna, <a href="/A379204/b379204.txt">Table of n, a(n) for n = 1..200</a>
%F A379204 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A379204 (1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).
%F A379204 (2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 4)^n.
%F A379204 (3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 4*A(x)^(n+1))^n.
%F A379204 (4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 4*A(x)^(n+1))^(n+1).
%F A379204 (5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 4)^(n+1) ).
%e A379204 G.f.: A(x) = x + 6*x^2 + 52*x^3 + 572*x^4 + 7154*x^5 + 96444*x^6 + 1365480*x^7 + 20015404*x^8 + 301104656*x^9 + 4622137698*x^10 + ...
%o A379204 (PARI) {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
%o A379204 V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 4)^(m+1) ), #V-3); ); polcoef(A, n)}
%o A379204 for(n=1, 40, print1(a(n), ", "))
%Y A379204 Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379205.
%K A379204 nonn
%O A379204 1,2
%A A379204 _Paul D. Hanna_, Dec 20 2024