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%I #21 Jun 12 2023 09:10:02
%S 1,25,1089,60025,3690241,241025625,16359689025,1140463805625,
%T 81081830657025,5852177325225625,427465780890020929,
%U 31528177440967935225,2344153069158724611841,175473167541934734763225,13211212029033949825064769,999630716942846408773325625
%N Expansion of g.f.: 1 / AGM(1-9*x, sqrt((1-x)*(1-81*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="/A246923/b246923.txt">Table of n, a(n) for n = 0..525</a>
%F a(n) = A084771(n)^2 = ( Sum_{k=0..n} 2^k*C(n,k)*C(2*k,k) )^2.
%F G.f.: 1 / AGM((1-x)*(1+9*x), (1+x)*(1-9*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F a(n) ~ 3^(4*n+2) / (8*Pi*n). - _Vaclav Kotesovec_, Sep 27 2019
%F a(n) = 9^n * ( LegendreP(n, 5/3) )^2. - _G. C. Greubel_, May 30 2023
%e G.f.: A(x) = 1 + 25*x + 1089*x^2 + 60025*x^3 + 3690241*x^4 + 241025625*x^5 +...
%e where the square-root of each term yields A084771:
%e [1, 5, 33, 245, 1921, 15525, 127905, 1067925, ...],
%e the g.f. of which is 1/sqrt((1-x)*(1-9*x)).
%t a[n_] := Sum[2^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 17, 0] (* _Amiram Eldar_, Dec 11 2018 *)
%t Table[9^n*LegendreP[n, 5/3]^2, {n, 0, 40}] (* _G. C. Greubel_, May 30 2023 *)
%o (PARI) {a(n,p=1,q=9)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n,p=1,q=9)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n,p=1,q=9)=sum(k=0,n,p^(n-k)*((q-p)/4)^k*binomial(n,k)*binomial(2*k,k))^2 }
%o for(n=0, 20, print1(a(n), ", "))
%o (Magma) [9^n*Evaluate(LegendrePolynomial(n), 5/3)^2 : n in [0..40]]; // _G. C. Greubel_, May 30 2023
%o (SageMath) [9^n*gen_legendre_P(n, 0, 5/3)^2 for n in range(41)] # _G. C. Greubel_, May 30 2023
%Y Cf. A084771, A246467, A246906, A248167.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 07 2014
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