A374497 - OEIS

%I #15 Oct 19 2024 08:31:19

%S 1,6,36,200,1080,5712,29792,153792,787680,4009280,20304768,102405888,

%T 514678528,2579028480,12890311680,64283809792,319954540032,

%U 1589720712192,7886437652480,39069462835200,193307835764736,955361266917376,4716674314223616,23264437702656000

%N Expansion of 1/(1 - 4*x - 4*x^2)^(3/2).

%F a(0) = 1, a(1) = 6; a(n) = (2*(2*n+1)*a(n-1) + 4*(n+1)*a(n-2))/n.

%F a(n) = binomial(n+2,2) * A071356(n).

%F a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(k,n-k). - _Seiichi Manyama_, Oct 19 2024

%o (PARI) a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1));

%Y Cf. A001788, A002457, A025163.

%Y Cf. A006139, A374511, A374513.

%Y Cf. A071356.

%K nonn,changed

%O 0,2

%A _Seiichi Manyama_, Jul 09 2024