OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
FORMULA
n*a(n) = 8*n*a(n-1) - 12*(n-1)*a(n-2) for n > 1.
a(n) = (1/2)^n * Sum_{k=0..n} 3^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^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
MATHEMATICA
CoefficientList[Series[Sqrt[(1-2*x)/(1-6*x)^3], {x, 0, 33}], x] (* Vincenzo Librandi, Aug 23 2025 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)^3))
(Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 2*x) / (1-6*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 22 2025
STATUS
approved