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A363569 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1.

%I A363569 #20 Aug 07 2023 12:51:04
%S A363569 1,0,1,-3,1,4,-4,-8,1,23,-8,-12,-27,12,36,15,1,16,-149,-20,71,91,166,
%T A363569 -24,-119,-186,-285,251,-26,28,761,-32,1,-199,-679,-827,-310,36,970,
%U A363569 1572,1821,40,-2631,-44,331,-5628,1772,-48,-495,3051,2546,6697,-1715,52,2791,-7425,-8007,-10869,-3653,-60
%N A363569 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1.
%C A363569 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds as a formal power series for all y.
%H A363569 Paul D. Hanna, <a href="/A363569/b363569.txt">Table of n, a(n) for n = 0..4100</a>
%F A363569 The g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas; here, i = (+/-) sqrt(-1).
%F A363569 (1) A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n).
%F A363569 (2) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + i*x^n)^(2*n).
%F A363569 (3) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) * (1 + i*x^n)^(2*n) / (1 + x^(2*n))^(2*n).
%F A363569 (4) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (i + x^n)^n.
%F A363569 (5) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + i*x^n)^n.
%F A363569 (6) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) * (1 + i*x^n)^n / (1 + x^(2*n))^n.
%F A363569 From _Paul D. Hanna_, Aug 06 2023: (Start)
%F A363569 The following generating functions are extensions of _Peter Bala_'s formulas given in A260147.
%F A363569 (7) A(x^2) = Sum_{n=-oo..+oo} x^(2*n+1) * (i + x^(2*n+1))^(2*n+1).
%F A363569 (8) A(x^2) = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + i*x^(2*n+1))^(2*n+1).
%F A363569 (End)
%F A363569 a(2^n) = 1 for n > 0 (conjecture).
%F A363569 a(p) = p*(-1)^((p-1)/2) - 1 for primes p > 3 (conjecture).
%e A363569 G.f.: A(x) = 1 + x^2 - 3*x^3 + x^4 + 4*x^5 - 4*x^6 - 8*x^7 + x^8 + 23*x^9 - 8*x^10 - 12*x^11 - 27*x^12 + 12*x^13 + 36*x^14 + 15*x^15 + ...
%o A363569 (PARI) {a(n) = my(A); A = sum(m=-n-1,n+1, x^m * (I + x^m +x*O(x^n))^(2*m) ); polcoeff(A,n)}
%o A363569 for(n=0,60,print1(a(n),", "))
%Y A363569 Cf. A260147, A363558, A363559.
%K A363569 sign
%O A363569 0,4
%A A363569 _Paul D. Hanna_, Jul 31 2023