A387237 Expansion of 1/((1-x) * (1-5*x))^(5/2).
1, 15, 145, 1155, 8260, 55188, 351960, 2170080, 13042095, 76827465, 445335891, 2547479025, 14412134100, 80773641900, 449065521300, 2479190589180, 13603361708775, 74238475926825, 403197150223175, 2180369322394725, 11744998515662720, 63044308615576200, 337323759106291100
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))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025 -
Mathematica
CoefficientList[Series[1/((1-x)*(1-5*x))^(5/2),{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))^(5/2))
Formula
n*a(n) = (6*n+9)*a(n-1) - 5*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k) = (binomial(n+4,2)/6) * A026377(n+2).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).