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Expansion of Gaussian product of arithmetic mean and Lehmer mean evaluated at 1 + 4*x.
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%I #14 May 06 2021 17:00:38

%S 1,2,2,-4,10,-28,76,-184,378,-620,708,-328,-36,-4008,25960,-83984,

%T 135066,188788,-2232236,9070040,-23335660,29933128,69732568,

%U -630906096,2551027132,-7193660232,13576199192,-5110303664,-92186317112

%N Expansion of Gaussian product of arithmetic mean and Lehmer mean evaluated at 1 + 4*x.

%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 265-266, Exercise 7.

%F G.f.: A(x) satisfies A(x) = (1 + 2*x) * A(x^2 / (1 + 2*x)^2), A(0) = 1.

%F a(n) = (-1)^n * Sum_{k>0} 2^(n - 2*k) * a(k) * binomial(n-2, n-2*k), n>1.

%e 1 + 2*x + 2*x^2 - 4*x^3 + 10*x^4 - 28*x^5 + 76*x^6 - 184*x^7 + ...

%o (PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=2; A = (1 + 2*x) * subst(A, x, x^2 / (1 + 2*x)^2)); polcoeff(A, n))}

%K sign,easy

%O 0,2

%A _Michael Somos_, Dec 05 2002