A383944 Expansion of 1/sqrt((1-7*x)^3 * (1+x)).
1, 10, 87, 708, 5565, 42798, 324275, 2430536, 18068409, 133454610, 980588367, 7174290060, 52301288949, 380120468406, 2755437681195, 19928252747664, 143839643441265, 1036380251867418, 7455465737930567, 53557027924956500, 384241833300244269, 2753539115904779070
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1-7*x)^3 * (1+x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025 -
Mathematica
CoefficientList[Series[1/Sqrt[(1-7*x)^3*(1+x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-7*x)^3*(1+x)))
Formula
n*a(n) = (6*n+4)*a(n-1) + 7*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).