OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
n*a(n) = (6*n-2)*a(n-1) - 5*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n+1,n-k).
MATHEMATICA
CoefficientList[Series[1/Sqrt[(1-x)^3*(1-5*x)], {x, 0, 33}], x] (* Vincenzo Librandi, Aug 27 2025 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-x)^3*(1-5*x)))
(Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- x)^3 * (1-5*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 19 2025
STATUS
approved