A385728 Expansion of 1/((1-2*x) * (1-6*x))^(3/2).

1, 12, 102, 760, 5310, 35784, 235788, 1530288, 9824310, 62557000, 395797908, 2491381776, 15616141996, 97537784400, 607391245080, 3772617319008, 23379854507046, 144605546475336, 892834113930180, 5504041611527760, 33883431379007364, 208327771987901808

Offset: 0

Author

Seiichi Manyama, Aug 19 2025

Keywords

• nonn
• new

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A385563, A385813.

Cf. A005572, A081671, A331515.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1 / ((1 - 2*x) * (1 - 6*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 22 2025

Mathematica

Module[{a, n}, RecurrenceTable[{a[n] == ((8*n+4)*a[n-1] - 12*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 12}, a, {n, 0, 25}]] (* Paolo Xausa, Aug 21 2025 *)
CoefficientList[Series[ 1/((1-2*x)*(1-6*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 22 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(3/2))

Formula

n*a(n) = (8*n+4)*a(n-1) - 12*(n+1)*a(n-2) for n > 1.

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

a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).

a(n) = binomial(n+2,2) * A005572(n).

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

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

a(n) ~ sqrt(n) * 2^(n - 1/2) * 3^(n + 3/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 21 2025