A387230 Expansion of sqrt((1-x) / (1-13*x)^5).
1, 32, 723, 14044, 250415, 4224732, 68565049, 1081299296, 16679767923, 252819395920, 3777709472537, 55782986878164, 815526073468561, 11821376147023268, 170096339292264375, 2431786467331116016, 34569517907583692867, 488963045591838160848, 6885041951078984405449
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) / (1-13*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025 -
Mathematica
CoefficientList[Series[Sqrt[(1-x)/(1-13*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(sqrt((1-x)/(1-13*x)^5))
Formula
n*a(n) = (14*n+18)*a(n-1) - 13*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^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} 3^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * n^(3/2) * 13^(n - 1/2) / sqrt(3*Pi). - Vaclav Kotesovec, Aug 24 2025