A387211 Expansion of sqrt((1-2*x) / (1-6*x)^3).
1, 8, 58, 400, 2678, 17584, 113892, 730272, 4646310, 29380912, 184867148, 1158418144, 7233806524, 45038743520, 279704675464, 1733203476288, 10718950211334, 66176597723184, 407931346057020, 2511127341708384, 15438601388617044, 94810212917983392, 581639541983344632
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 2*x) / (1-6*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025 -
Mathematica
CoefficientList[Series[Sqrt[(1-2*x)/(1-6*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)^3))
Formula
n*a(n) = 8*n*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) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 2^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).