A387210 Expansion of sqrt((1-x) / (1-13*x)^3).
1, 19, 307, 4645, 67843, 969337, 13643533, 189953659, 2622877075, 35982412921, 491057325577, 6672763735183, 90347244052429, 1219537191931975, 16418449380961891, 220534056531679141, 2956293832279184659, 39559312793250153577, 528522358385088314425, 7051193680459915645903
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) / (1-13*x)^3)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025 -
Mathematica
CoefficientList[Series[Sqrt[(1-x)/(1-13*x)^3],{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)^3))
Formula
n*a(n) = (14*n+5)*a(n-1) - 13*(n-1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^k * (2*k+1) * 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) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
a(n) ~ 4 * sqrt(3*n) * 13^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 23 2025