A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

1, 2, -1, 3, -10, 36, -137, 543, -2219, 9285, -39587, 171369, -751236, 3328218, -14878455, 67030785, -304036170, 1387247580, -6363044315, 29323149825, -135700543190, 630375241380, -2938391049395, 13739779184085, -64430797069375, 302934667061301, -1427763630578197

Offset

0,2

Links

Table of n, a(n) for n=0..26.

Formula

G.f.: A(x) = -2*x*(1+x) / (1+x-sqrt((1+x)*(1+5*x))).

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

a(n) ~ -(-1)^n * 5^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 07 2023

From Peter Bala, Sep 10 2024: (Start)

a(n) = 1/(1 - n) * Sum_{k = 0..n} binomial(-n+k, k)*binomial(-n+k+1, n-k) for n not equal to 1. Cf. A007863.

a(n) = Sum_{k = 0..n-2} binomial(-n+k+1, k)*binomial(-n+k+1, n-k)/(-n+k+1) for n >= 2.

P-recursive: n*a(n) = - 3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2) with a(1) = 2 and a(2) = -1. (End)

Maple

a := proc(n) option remember; if n = 1 then 2 elif n = 2 then -1 else (-3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2))/n fi; end:
seq(a(n), n = 1..30); # Peter Bala, Sep 10 2024

Prog

(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(n-2, n-k)/(2*k-1));

Crossrefs

Cf. A073157, A007863, A364336, A364337, A364338, A364339, A366326, A366327, A366328.

Cf. A000108, A002212, A112478.

Sequence in context: A070165 A192719 A270996 * A203709 A270382 A340063

Adjacent sequences: A366322 A366323 A366324 * A366326 A366327 A366328

Keyword

sign

Author

Seiichi Manyama, Oct 07 2023

Status

approved