OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..800
FORMULA
n*a(n) = (6*n+11)*a(n-1) + 7*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * (-1)^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * binomial(2*k,k) * binomial(n+2,n-k).
MATHEMATICA
CoefficientList[Series[1/Sqrt[(1-7*x)^5*(1+x)], {x, 0, 33}], x] (* Vincenzo Librandi, Aug 27 2025 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-7*x)^5*(1+x)))
(Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1-7*x)^5 * (1+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
