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A363558 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).

%I A363558 #17 Aug 07 2023 08:04:01
%S A363558 1,5,16,77,256,1104,4121,16832,65536,264688,1048617,4205568,16779008,
%T A363558 67162112,268436016,1073999165,4294967296,17180983296,68719549696,
%U A363558 274882887680,1099511628896,4398068904021,17592186086656,70368840646656,281474978676736,1125900326286464
%N A363558 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).
%C A363558 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds as a formal power series for all y.
%H A363558 Paul D. Hanna, <a href="/A363558/b363558.txt">Table of n, a(n) for n = 0..300</a>
%F A363558 The g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
%F A363558 (1.a) A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).
%F A363558 (1.b) A(x) = Sum_{n=-oo..+oo} x^n * (2 - x^n)^(2*n).
%F A363558 (2.a) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - 2*x^n)^(2*n).
%F A363558 (2.b) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + 2*x^n)^(2*n).
%F A363558 (3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (2 + x^n)^n.
%F A363558 (3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (-2 + x^n)^n.
%F A363558 (4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + 2*x^n)^n.
%F A363558 (4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - 2*x^n)^n.
%F A363558 From _Paul D. Hanna_, Aug 06 2023: (Start)
%F A363558 The following generating functions are extensions of _Peter Bala_'s formulas given in A260147.
%F A363558 (5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n+1) * (2 + x^(2*n+1))^(2*n+1).
%F A363558 (5.b) A(x^2) = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + 2*x^(2*n+1))^(2*n+1).
%F A363558 (End)
%F A363558 a(2^n) = 4^(2^n) for n > 0 (conjecture).
%F A363558 a(p) = p*2^(p-1) + 4^p for primes p > 3 (conjecture).
%e A363558 G.f.: A(x) = 1 + 5*x + 16*x^2 + 77*x^3 + 256*x^4 + 1104*x^5 + 4121*x^6 + 16832*x^7 + 65536*x^8 + 264688*x^9 + 1048617*x^10 + ...
%o A363558 (PARI) {a(n) = my(A); A = sum(m=-n-1,n+1, x^m * (2 + x^m +x*O(x^n))^(2*m) ); polcoeff(A,n)}
%o A363558 for(n=0,30,print1(a(n),", "))
%Y A363558 Cf. A260147, A363559, A363569, A363561.
%K A363558 nonn
%O A363558 0,2
%A A363558 _Paul D. Hanna_, Aug 01 2023