A265610 a(n) = rf(n, n+2)/(n+2)! - rf(n, n)/n!, rf the rising factorial A265609.

-1, 0, 2, 11, 49, 204, 825, 3289, 13013, 51272, 201552, 791350, 3105322, 12183560, 47805615, 187623765, 736618125, 2893125840, 11367801060, 44686512090, 175739405790, 691437981000, 2721606268290, 10717182330426, 42219554975874, 166386610183024, 655976895434000

Offset: 0

Author

Peter Luschny, Dec 19 2015

Keywords

• sign

Crossrefs

Cf. A002054, A088218.

Programs

Magma

[Binomial(2*n+1, n-1)-(0^n + Binomial(2*n, n))/2: n in [0..30]]; // Vincenzo Librandi, Dec 20 2015

Mathematica

Join[{-1}, Table[Binomial[2 n + 1, n - 1] - Binomial[2 n, n]/2, {n, 1, 36}]] (* Vincenzo Librandi, Dec 20 2015 *)

Sage

A265610 = lambda n: rising_factorial(n, n+2)/factorial(n+2) - rising_factorial(n, n)/factorial(n)
print([A265610(n) for n in srange(27)])

Formula

G.f.: (-x^2+x-1-(x^2+3*x-1)/sqrt(1-4*x))/(2*x^2).

a(n) = binomial(2*n+1, n-1)-(0^n + binomial(2*n, n))/2 = A002054(n) - A088218(n).

a(n) = (1/2)*(3*n+2)*(n+2)*(n^2-1)*Gamma(2*n+1)/Gamma(n+3)^2 for n>=1.

a(n) ~ 4^n*(3/2)/sqrt(n*Pi).