This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A246906 #18 Sep 27 2019 16:50:58
%S A246906 1,25,729,24025,866761,33350625,1342856025,55849505625,2378365418025,
%T A246906 103099146750625,4531090723144129,201324497403240225,
%U A246906 9025111586043157801,407581475160408424225,18521763259935613598649,846187436813348419025625,38838031986984135802130025
%N A246906 G.f.: 1 / AGM(1-21*x, sqrt((1-9*x)*(1-49*x))).
%C A246906 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 A246906 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 A246906 and consists of integer coefficients when 4|(q-p).
%C A246906 Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
%H A246906 Seiichi Manyama, <a href="/A246906/b246906.txt">Table of n, a(n) for n = 0..593</a>
%F A246906 a(n) = A098409(n)^2 = [Sum_{k=0..n} 3^(n-k)*C(n,k)*C(2*k,k)]^2.
%F A246906 G.f.: 1 / AGM((1-3*x)*(1+7*x), (1+3*x)*(1-7*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F A246906 From _Vaclav Kotesovec_, Sep 27 2019: (Start)
%F A246906 Recurrence: n^2*(2*n - 3)*a(n) = (2*n - 1)*(79*n^2 - 158*n + 54)*a(n-1) - 21*(2*n - 3)*(79*n^2 - 158*n + 54)*a(n-2) + 9261*(n-2)^2*(2*n - 1)*a(n-3).
%F A246906 a(n) ~ 7^(2*n+1) / (4*Pi*n). (End)
%e A246906 G.f.: A(x) = 1 + 25*x + 729*x^2 + 24025*x^3 + 866761*x^4 +...
%e A246906 where the square-root of the terms yields A098409:
%e A246906 [1, 5, 27, 155, 931, 5775, 36645, 236325, 1542195, ...],
%e A246906 the g.f. of which is 1/sqrt((1-3*x)*(1-7*x)).
%t A246906 a[n_] := Sum[3^(n - k) * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 17, 0] (* _Amiram Eldar_, Dec 11 2018 *)
%o A246906 (PARI) {a(n,p=3,q=7)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
%o A246906 for(n=0, 20, print1(a(n), ", "))
%o A246906 (PARI) {a(n,p=3,q=7)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
%o A246906 for(n=0, 20, print1(a(n), ", "))
%o A246906 (PARI) {a(n,p=3,q=7)=sum(k=0,n,p^(n-k)*((q-p)/4)^k*binomial(n,k)*binomial(2*k,k))^2 }
%o A246906 for(n=0, 20, print1(a(n), ", "))
%Y A246906 Cf. A098409, A246467, A248167.
%K A246906 nonn
%O A246906 0,2
%A A246906 _Paul D. Hanna_, Sep 07 2014