OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{k=0..n} (-3)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).
a(n) = Sum_{k=0..n} (-2)^k * (k+1) * binomial(n+1,k+1)^2.
a(n) = (n + 1)^2*hypergeom([-n, -n], [2], -2). - Peter Luschny, Jan 20 2020
n * (2*n-1) * a(n) = 4 * (-n^2 + 1) * a(n-1) - 9 * n * (2*n+1) * a(n-2) for n>1. - Seiichi Manyama, Jan 25 2020
MAPLE
a := n -> (n + 1)^2*hypergeom([-n, -n], [2], -2):
seq(simplify(a(n)), n=0..19); # Peter Luschny, Jan 20 2020
MATHEMATICA
a[n_] := Sum[(-2)^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 25, 0] (* Amiram Eldar, Jan 20 2020 *)
PROG
(PARI) N=20; x='x+O('x^N); Vec((1+3*x)/(1+2*x+9*x^2)^(3/2))
(PARI) {a(n) = sum(k=0, n, (-3)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
(PARI) {a(n) = sum(k=0, n, (-2)^k*(k+1)*binomial(n+1, k+1)^2)}
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 + 3*x)/(1 + 2*x + 9*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jan 20 2020
STATUS
approved