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%I #35 Dec 10 2018 08:51:48
%S 1,16,324,7744,206116,5875776,175191696,5386385664,169300977444,
%T 5410164352576,175128910042384,5727842622630144,188931648862083856,
%U 6276176070222305536,209747841324097564224,7046053064278540084224,237764385841359952067364,8054915184317632144620096
%N G.f.: 1 / AGM(1-12*x, sqrt((1-4*x)*(1-36*x))).
%C In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by
%C 1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*[Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k)]^2,
%C and consists of integer coefficients when 4|(q-p).
%C Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
%H Seiichi Manyama, <a href="/A246876/b246876.txt">Table of n, a(n) for n = 0..644</a>
%F a(n) = A081671(n)^2 = [Sum_{k=0..n} 2^(n-k) * C(n,k) * C(2*k,k)]^2.
%F G.f.: 1 / AGM((1-2*x)*(1+6*x), (1+2*x)*(1-6*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F a(n) ~ 2^(2*n - 1) * 3^(2*n + 1) / (Pi*n). - _Vaclav Kotesovec_, Dec 10 2018
%e G.f.: A(x) = 1 + 16*x + 324*x^2 + 7744*x^3 + 206116*x^4 + 5875776*x^5 +...
%e where the square-root of the terms yields A081671:
%e [1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, ...]
%e the g.f. of which is 1/sqrt((1-2*x)*(1-6*x)).
%o (PARI) {a(n)=polcoeff( 1 / agm(1-12*x, sqrt((1-4*x)*(1-36*x) +x*O(x^n))), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=sum(k=0,n,2^(n-k)*binomial(n,k)*binomial(2*k,k))^2}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A081671, A168597, A246467.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 06 2014
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