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 A322249 #8 Dec 10 2018 08:49:23
%S A322249 1,25,3249,366025,48455521,6646325625,947789867025,138422872355625,
%T A322249 20598606105401025,3109408600719825625,474780862425986767729,
%U A322249 73175222677868396505225,11366022325041154078402081,1777059915791491092441607225,279404859303904515406536763089,44144451113336819125597110875625,7004264626817806908496520658161025,1115512654966236899680358546064905625
%N A322249 a(n) = A322248(n)^2, the square of the central coefficient in (1 + 5*x + 16*x^2)^n.
%H A322249 Seiichi Manyama, <a href="/A322249/b322249.txt">Table of n, a(n) for n = 0..450</a>
%F A322249 G.f.: 1 / AGM(1 + 3*13*x, sqrt((1 - 3^2*x)*(1 - 13^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
%F A322249 G.f.: 1 / AGM((1-3*x)*(1-13*x), (1+3*x)*(1+13*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F A322249 a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
%F A322249 a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
%F A322249 a(n) ~ 13^(2*n + 1) / (16*Pi*n). - _Vaclav Kotesovec_, Dec 10 2018
%e A322249 G.f.: A(x) = 1 + 25*x + 3249*x^2 + 366025*x^3 + 48455521*x^4 + 6646325625*x^5 + 947789867025*x^6 + 138422872355625*x^7 + 20598606105401025*x^8 + ...
%e A322249 such that
%e A322249 A(x) = 1 + 5^2*x + 57^2*x^2 + 605^2*x^3 + 6961^2*x^4 + 81525^2*x^5 + 973545^2*x^6 + 11765325^2*x^7 + 143522145^2*x^8 + ... + A322248(n)^2*x^n + ...
%o A322249 (PARI) /* a(n) = A322248(n)^2 - g.f. */
%o A322249 {a(n)=polcoeff(1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)), n)^2}
%o A322249 for(n=0, 20, print1(a(n), ", "))
%o A322249 (PARI) /* a(n) = A322248(n)^2 - g.f. */
%o A322249 {a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)^2}
%o A322249 for(n=0, 20, print1(a(n), ", "))
%o A322249 (PARI) /* a(n) = A322248(n)^2 - binomial sum */
%o A322249 {a(n) = sum(k=0,n, (-3)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))^2}
%o A322249 for(n=0, 20, print1(a(n), ", "))
%o A322249 (PARI) /* Using binomial formula: */
%o A322249 {a(n) = sum(k=0,n, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))^2}
%o A322249 for(n=0,30,print1(a(n),", "))
%o A322249 (PARI) /* Using AGM: */
%o A322249 {a(n)=polcoeff( 1 / agm(1 + 3*13*x, sqrt((1 - 3^2*x)*(1 - 13^2*x) +x*O(x^n))), n)}
%o A322249 for(n=0, 20, print1(a(n), ", "))
%Y A322249 Cf. A246467, A246906, A246923, A322248.
%K A322249 nonn
%O A322249 0,2
%A A322249 _Paul D. Hanna_, Dec 10 2018