A387366 Expansion of 1/(1 - 10*x + x^2)^(3/2).
1, 15, 186, 2150, 23955, 260925, 2798740, 29688300, 312289605, 3263403275, 33922215822, 351081270930, 3620347505047, 37217828876025, 381591426746280, 3903412392243800, 39848499404096265, 406072116038615175, 4131456665470332130, 41974347760312761150, 425899035044461953051
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/(1 - 10*x + x^2)^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 29 2025 -
Mathematica
CoefficientList[Series[1/(1-10*x+x^2)^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 29 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/(1-10*x+x^2)^(3/2))
Formula
n*a(n) = 5*(2*n+1)*a(n-1) - (n+1)*a(n-2) for n > 1.
a(n) = ((n+2)/2) * A387368(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/10)^(n-2*k) * binomial(-3/2,k) * binomial(k,n-k).