login
A360317
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
4
1, 2, 10, 52, 278, 1516, 8388, 46920, 264678, 1503052, 8581676, 49215256, 283297660, 1635904376, 9472214344, 54975423504, 319729353606, 1862896455180, 10871759717916, 63539265366264, 371837338366740, 2178604586281128, 12778264475444280, 75022726995053808
OFFSET
0,2
FORMULA
G.f.: sqrt( (1-2*x)/(1-6*x) ).
n*a(n) = 2*(4*n-3)*a(n-1) - 12*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/2)^i * a(j) * a(i-j) = 3^n.
a(n) = 2 * A005573(n-1) for n > 0.
a(n) ~ 2^(n + 1/2) * 3^(n - 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
PROG
(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
(PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)))
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Feb 03 2023
STATUS
approved