A387343 Expansion of 1/(1 - 8*x + 4*x^2)^(5/2).
1, 20, 270, 3080, 31990, 312984, 2937900, 26751120, 237977190, 2078447800, 17884238372, 152002796400, 1278603975740, 10660760170480, 88213513627800, 725107271106336, 5925674432448390, 48175954959638520, 389871795632108020, 3142078444590396080, 25228464363569709396
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/(1 - 8*x + 4*x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025 -
Mathematica
CoefficientList[Series[1/(1-8*x+4*x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(5/2))
Formula
n*a(n) = 4*(2*n+3)*a(n-1) - 4*(n+3)*a(n-2) for n > 1.
a(n) = (binomial(n+4,2)/6) * A387339(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/2)^(n-4*k) * binomial(-5/2,k) * binomial(k,n-k).