A383945 Expansion of 1/sqrt((1-7*x)^5 * (1+x)).
1, 17, 206, 2150, 20615, 187103, 1633996, 13868508, 115147965, 939490365, 7557020922, 60073436514, 472815344547, 3689827880235, 28584232842840, 220017882647544, 1683964821974073, 12824134005685929, 97224403777732070, 734127854369080990, 5523136813883811199
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)^5 * (1+x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025 -
Mathematica
CoefficientList[Series[1/Sqrt[(1-7*x)^5*(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)^5*(1+x)))
Formula
n*a(n) = (6*n+11)*a(n-1) + 7*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * (-1)^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(2*k,k) * binomial(n+2,n-k).