A360319 - OEIS

%I #15 Feb 04 2023 10:19:49

%S 1,2,14,100,726,5340,39692,297544,2245990,17050796,130061412,

%T 996078456,7654571772,58995989400,455857911768,3530234227344,

%U 27392392806534,212918339726028,1657570714812020,12922254685161112,100867892292766612

%N a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

%F G.f.: sqrt( (1-4*x)/(1-8*x) ).

%F n*a(n) = 2*(6*n-5)*a(n-1) - 32*(n-2)*a(n-2).

%F Sum_{i=0..n} Sum_{j=0..i} (1/4)^i * a(j) * a(i-j) = 2^n.

%F a(n) ~ 2^(3*n - 1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 04 2023

%o (PARI) a(n) = sum(k=0, n, 4^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-4*x)/(1-8*x)))

%Y Cf. A063886, A085362, A360317, A360318, A360321, A360322.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Feb 03 2023