A387208 Expansion of sqrt((1-x) / (1-9*x)^3).
1, 13, 145, 1517, 15329, 151565, 1476465, 14228205, 135990465, 1291409165, 12199991633, 114761111789, 1075651464865, 10051341904141, 93677905064497, 871083359663085, 8083754402585985, 74885500462111245, 692624008942816785, 6397104350057979885, 59008673876627412321
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 34); f := Sqrt((1-x) / (1-9*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025 -
Mathematica
CoefficientList[Series[Sqrt[(1-x)/(1-9*x)^3],{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)^3))
Formula
n*a(n) = (10*n+3)*a(n-1) - 9*(n-1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * 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) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
a(n) ~ 2^(5/2) * sqrt(n) * 3^(2*n-1) / sqrt(Pi). - Vaclav Kotesovec, Aug 23 2025