A360319 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
1, 2, 14, 100, 726, 5340, 39692, 297544, 2245990, 17050796, 130061412, 996078456, 7654571772, 58995989400, 455857911768, 3530234227344, 27392392806534, 212918339726028, 1657570714812020, 12922254685161112, 100867892292766612
Offset: 0
Programs
-
PARI
a(n) = sum(k=0, n, 4^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
-
PARI
my(N=30, x='x+O('x^N)); Vec(sqrt((1-4*x)/(1-8*x)))
Formula
G.f.: sqrt( (1-4*x)/(1-8*x) ).
n*a(n) = 2*(6*n-5)*a(n-1) - 32*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/4)^i * a(j) * a(i-j) = 2^n.
a(n) ~ 2^(3*n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 8^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)