A331515 Expansion of 1/(1 - 8*x + 4*x^2)^(3/2).

1, 12, 114, 1000, 8430, 69384, 561988, 4499856, 35719830, 281634760, 2208564732, 17242680624, 134118558028, 1039939550160, 8041848166920, 62042202765856, 477670318108902, 3670988584476744, 28166853684793420, 215807899372086000, 1651323989374972836

Offset: 0

Author

Seiichi Manyama, Jan 19 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column 4 of A331514.

Cf. A007564, A069835, A385728.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 21); Coefficients(R!( 1/(1 - 8*x + 4*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020

Magma

[&+[2^(n-k)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..21]]; // Marius A. Burtea, Jan 20 2020

Mathematica

a[n_] := Sum[2^(n - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n + 1}]; Array[a, 21, 0] (* Amiram Eldar, Jan 20 2020 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(3/2))

PARI

a(n) = sum(k=1, n+1, 2^(n-k)*k*binomial(n+1, k)*binomial(n+1+k, k));

Formula

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

n * a(n) = 4 * (2*n+1) * a(n-1) - 4 * (n+1) * a(n-2) for n>1.

a(n) = ((n+2)/2) * Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).

a(n) ~ 2^(n - 1/2) * (2 + sqrt(3))^(n + 3/2) * sqrt(n) / (3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020

From Seiichi Manyama, Aug 20 2025: (Start)

a(n) = binomial(n+2,2) * A007564(n+1).

a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).

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