A387212 Expansion of sqrt((1-3*x) / (1-7*x)^3).
1, 9, 75, 599, 4659, 35595, 268485, 2005785, 14873715, 109643195, 804354417, 5877232773, 42798735805, 310767250773, 2250899498763, 16267896905895, 117347641620435, 845043416086635, 6076092412278465, 43629213402099045, 312892629725930121, 2241442380182752209
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- 3*x) / (1-7*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025 -
Mathematica
CoefficientList[Series[Sqrt[(1-3*x)/(1-7*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)^3))
Formula
n*a(n) = (10*n-1)*a(n-1) - 21*(n-1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).