A387228 Expansion of sqrt((1-x) / (1-5*x)^5).
1, 12, 103, 764, 5215, 33728, 210021, 1271504, 7532547, 43859460, 251809701, 1428911652, 8028877233, 44734340784, 247433518875, 1359902816880, 7432212863235, 40416897046740, 218812616979845, 1179889937796900, 6339243523221245, 33947223885549040, 181245459484155935
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 := Sqrt((1- x) / (1-5*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025 -
Mathematica
CoefficientList[Series[Sqrt[(1-x)/(1-5*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-5*x)^5))
Formula
n*a(n) = (6*n+6)*a(n-1) - 5*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * 5^(n - 1/2) * n^(3/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, Aug 23 2025