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%I #15 Sep 27 2019 17:23:30
%S 1,9,289,9801,370881,14768649,609546721,25795893321,1112301387649,
%T 48661046208009,2153525838773409,96206353829163081,
%U 4331637064535243841,196320612369490652169,8948100956521251936609,409841929215895450531401,18851871634769751620818689,870412126485504031282344969,40322236319029726815932366881,1873525179289326411511891685961,87284881539374630658475997415361
%N a(n) = A322242(n)^2, the square of the central coefficient in (1 + 3*x + 4x^2)^n.
%C The g.f. of A322242 is 1/sqrt(1 - 6*x - 7*x^2).
%H Robert Israel, <a href="/A322243/b322243.txt">Table of n, a(n) for n = 0..593</a>
%F G.f.: 1 / AGM(1 + 7*x, sqrt((1 - x)*(1 - 49*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
%F G.f.: 1 / AGM((1-x)*(1-7*x), (1+x)*(1+7*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F a(n) = A322242(n)^2 where A322242(n) = Sum_{k=0..n} (-1)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k).
%F 343*(5+2*n)*(n+1)^2*a(n)-7*(3+2*n)*(43*n^2+172*n+163)*a(n+1)-(5+2*n)*(43*n^2+172*n+163)*a(n+2)+(3+2*n)*(n+3)^2*a(n+3)=0. - _Robert Israel_, Dec 10 2018
%F a(n) ~ 7^(2*n + 1) / (8*Pi*n). - _Vaclav Kotesovec_, Sep 27 2019
%e G.f.: A(x) = 1 + 9*x + 289*x^2 + 9801*x^3 + 370881*x^4 + 14768649*x^5 + 609546721*x^6 + 25795893321*x^7 + 1112301387649*x^8 + 48661046208009*x^9 + ...
%e such that
%e A(x) = 1 + 3^2*x + 17^2*x^2 + 99^2*x^3 + 609^2*x^4 + 3843^2*x^5 + 24689^2*x^6 + 160611^2*x^7 + 1054657^2*x^8 + 6975747^2*x^9 + ... + A322242(n)^2*x^n + ...
%p f:= gfun:-rectoproc({343*(5+2*n)*(n+1)^2*a(n)-7*(3+2*n)*(43*n^2+172*n+163)*a(n+1)-(5+2*n)*(43*n^2+172*n+163)*a(n+2)+(3+2*n)*(n+3)^2*a(n+3)=0, a(0)=1, a(1)=3^2, a(2)=17^2},a(n),remember):
%p map(f, [$0..30]); # _Robert Israel_, Dec 10 2018
%t f[n_] := (CoefficientList[Expand[(1 + 3*x + 4*x^2)^n], x][[n + 1]])^2; Array[f, 22, 0] (* _Amiram Eldar_, Dec 10 2018 *)
%t CoefficientList[Series[2*EllipticK[1 - (1 + 7*x)^2/((1 - 49*x)*(1 - x))] / (Pi*Sqrt[(1 - 49*x)*(1 - x)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 27 2019 *)
%o (PARI) /* a(n) = A322242(n)^2 */
%o {a(n)=polcoeff(1/sqrt(1 - 6*x - 7*x^2 +x*O(x^n)), n)^2}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* Using AGM: */
%o {a(n)=polcoeff( 1 / 1 / agm(1 + 7*x, sqrt((1 - x)*(1 - 7^2*x) +x*O(x^n))), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A322242.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 08 2018
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