A337370 - OEIS

%I #23 Aug 31 2020 02:57:19

%S 1,8,74,736,7606,80464,864772,9400192,103061158,1137528688,

%T 12623082284,140697113792,1574005263676,17663830073504,

%U 198760191043784,2241743315230208,25335473017856774,286850379192127664,3252960763923781276,36942512756224955456,420084161646913792724

%N Expansion of sqrt(2 / ( (1-12*x+4*x^2) * (1-2*x+sqrt(1-12*x+4*x^2)) )).

%H Robert Israel, <a href="/A337370/b337370.txt">Table of n, a(n) for n = 0..938</a>

%F a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).

%F 8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0. - _Robert Israel_, Aug 27 2020

%F a(0) = 1, a(1) = 8 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * (24*n^2-12*n-4) * a(n-1) - 4 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - _Seiichi Manyama_, Aug 29 2020

%F a(n) ~ 2^(n - 5/4) * (1 + sqrt(2))^(2*n + 3/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Aug 31 2020

%p Rec:= 8*(2*n - 3)*(n - 2)*a(n - 3) - 4*(10*n^2 - 35*n + 27)*a(n - 2) - 2*(10*n^2 + 5*n - 3)*a(n - 1) + (2*n + 1)*n*a(n) = 0:

%p f:= gfun:-rectoproc({Rec,a(0)=1,a(1)=8,a(2)=74},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Aug 27 2020

%t a[n_] := Sum[2^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Aug 25 2020 *)

%o (PARI) N=40; x='x+O('x^N); Vec(sqrt(2/((1-12*x+4*x^2)*(1-2*x+sqrt(1-12*x+4*x^2)))))

%o (PARI) {a(n) = sum(k=0, n, 2^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

%Y Column k=2 of A337369.

%Y Cf. A337390.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 25 2020