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A363561 G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).

%I A363561 #17 Aug 07 2023 08:03:56
%S A363561 1,3,4,15,16,52,77,184,256,716,1045,2400,4320,9024,16524,35439,65536,
%T A363561 135424,264928,534016,1048856,2124523,4196944,8435712,16792576,
%U A363561 33658512,67118016,134478584,268435513,537346048,1073876144,2148499456,4294967296,8592337520,17179956224,34364358760
%N A363561 G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).
%C A363561 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds as a formal power series for all y.
%H A363561 Paul D. Hanna, <a href="/A363561/b363561.txt">Table of n, a(n) for n = 0..1030</a>
%F A363561 The g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
%F A363561 (1.a) A(x) = Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).
%F A363561 (1.b) A(x) = Sum_{n=-oo..+oo} x^n * (sqrt(2) - x^n)^(2*n).
%F A363561 (2.a) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - sqrt(2)*x^n)^(2*n).
%F A363561 (2.b) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + sqrt(2)*x^n)^(2*n).
%F A363561 (3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^n.
%F A363561 (3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (-sqrt(2) + x^n)^n.
%F A363561 (4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + sqrt(2)*x^n)^n.
%F A363561 (4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - sqrt(2)*x^n)^n.
%F A363561 From _Paul D. Hanna_, Aug 06 2023: (Start)
%F A363561 The following generating functions are extensions of _Peter Bala_'s formulas given in A260147.
%F A363561 (5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n+1) * (sqrt(2) + x^(2*n+1))^(2*n+1).
%F A363561 (5.b) A(x^2) = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + sqrt(2)*x^(2*n+1))^(2*n+1).
%F A363561 (End)
%F A363561 a(2^n) = 2^(2^n) for n > 0 (conjecture).
%F A363561 a(p) = p*2^((p-1)/2) + 2^p for primes p > 3 (conjecture).
%e A363561 G.f.: A(x) = 1 + 3*x + 4*x^2 + 15*x^3 + 16*x^4 + 52*x^5 + 77*x^6 + 184*x^7 + 256*x^8 + 716*x^9 + 1045*x^10 + 2400*x^11 + 4320*x^12 + ...
%o A363561 (PARI) {a(n) = my(A); A = sum(m=-n-1, n+1, x^m * (sqrt(2) + x^m +x*O(x^n))^(2*m) ); polcoeff(A, n)}
%o A363561 for(n=0, 30, print1(round(a(n)), ", "))
%Y A363561 Cf. A260147, A363558, A363559, A363569.
%K A363561 nonn
%O A363561 0,2
%A A363561 _Paul D. Hanna_, Aug 01 2023