A387238 Expansion of 1/((1-x) * (1-5*x))^(7/2).
1, 21, 266, 2646, 22806, 178794, 1310694, 9140274, 61330269, 399107709, 2533330800, 15751925280, 96257031780, 579556206180, 3445117599480, 20252115155160, 117890464642335, 680320688005035, 3895668955041710, 22152779612619810, 125183331416173030
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 := 1/((1-x) * (1-5*x))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025 -
Mathematica
CoefficientList[Series[1/((1-x)*(1-5*x))^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(7/2))
Formula
n*a(n) = (6*n+15)*a(n-1) - 5*(n+5)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387239(n).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).