A383950 Expansion of 1/sqrt((1-2*x)^3 * (1-6*x)).
1, 6, 30, 148, 750, 3924, 21084, 115560, 642582, 3611140, 20455908, 116594328, 667851340, 3840932424, 22164538680, 128269848528, 744150592998, 4326419433060, 25200835078164, 147036927946680, 859181709840804, 5027183713857624, 29450272491511560, 172715082105669552
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- 2*x)^3 * (1-6*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025 -
Mathematica
CoefficientList[Series[1/Sqrt[(1-2*x)^3*(1-6*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-2*x)^3*(1-6*x)))
Formula
n*a(n) = (8*n-2)*a(n-1) - 12*n*a(n-2) for n > 1.
a(n) = (1/2)^n * Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).