A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

1, 2, -4, 10, -30, 102, -376, 1462, -5900, 24470, -103644, 446382, -1948854, 8605290, -38362200, 172423770, -780496110, 3554991270, -16281079900, 74927379550, -346328465930, 1607078948690, -7483861047480, 34963419415650, -163825013554400, 769694347677002

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Formula

G.f.: sqrt( (1+5*x)/(1+x) ).

n*a(n) = 2*(-3*n+4)*a(n-1) - 5*(n-2)*a(n-2).

Sum_{i=0..n} Sum_{j=0..i} (-1/5)^i * a(j) * a(i-j) = (1/5)^n.

a(n) = 2 * (-1)^(n+1) * A007317(n) for n > 0.

From Seiichi Manyama, Aug 22 2025: (Start)

a(n) = (-1/4)^n * Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).

a(n) = (-1)^n * Sum_{k=0..n} binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

Prog

(PARI) a(n) = sum(k=0, n, (-5)^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
(PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1+5*x)/(1+x)))

Crossrefs

Cf. A063886, A085362, A360317, A360318, A360319, A360321.

Cf. A007317.

Sequence in context: A000733 A092073 A274232 * A112846 A050397 A186021

Adjacent sequences: A360319 A360320 A360321 * A360323 A360324 A360325

Keyword

sign,easy

Author

Seiichi Manyama, Feb 03 2023

Status

approved