A331516 Expansion of 1/(1 - 10*x + 9*x^2)^(3/2).

1, 15, 174, 1850, 18855, 187425, 1832460, 17705700, 169569405, 1612842275, 15256106778, 143660483070, 1347716324227, 12603114069525, 117536416879320, 1093553079352200, 10153324144411065, 94098595671581175, 870667876141568070, 8044341506669534850

Offset: 0

Author

Seiichi Manyama, Jan 19 2020

Keywords

• nonn
• changed

Links

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

Crossrefs

Column 5 of A331514.

Cf. A059231, A084771, A383946.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(1 - 10*x + 9*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020

Magma

[1/2*&+[3^(n-k+1)*k*Binomial(n+1, k)*Binomial(n+k+1,k):k in [1..n+1]]:n in [0..20]]; // Marius A. Burtea, Jan 20 2020

Mathematica

a[n_] := 1/2 * Sum[3^( n + 1 - k) * k * Binomial[n + 1, k] * Binomial[n + 1 + k, k], {k, 1, n+1}]; Array[a, 20, 0] (* Amiram Eldar, Jan 20 2020 *)
CoefficientList[Series[1/(1-10x+9x^2)^(3/2),{x,0,20}],x] (* Harvey P. Dale, Nov 04 2021 *)

PARI

my(N=20, x='x+O('x^N)); Vec(1/(1-10*x+9*x^2)^(3/2))

PARI

a(n) = sum(k=1, n+1, 3^(n+1-k)*k*binomial(n+1, k)*binomial(n+1+k, k))/2;

Formula

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

n * a(n) = 5 * (2*n+1) * a(n-1) - 9 * (n+1) * a(n-2) for n > 1.

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

a(n) ~ sqrt(n) * 3^(2*n + 3) / (2^(7/2) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020

From Seiichi Manyama, Aug 20 2025: (Start)

a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).

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

a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).

a(n) = binomial(n+2,2) * A059231(n+1).

a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).

a(n) = Sum_{k=0..n} (5/2)^k * (-9/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)