A387229 Expansion of sqrt((1-x) / (1-9*x)^5).
1, 22, 343, 4604, 56765, 662450, 7438515, 81174840, 866564025, 9090485390, 94014360143, 960890353076, 9723664642549, 97564323687082, 971756818248235, 9616894723897200, 94635806917660785, 926607762721058310, 9032093873432341575, 87685949210949054060, 848182216775168898861
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) / (1-9*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025 -
Mathematica
CoefficientList[Series[Sqrt[(1-x)/(1-9*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-9*x)^5))
Formula
n*a(n) = (10*n+12)*a(n-1) - 9*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^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 * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 2^(7/2) * n^(3/2) * 3^(2*n-2) / sqrt(Pi). - Vaclav Kotesovec, Aug 24 2025