cp's OEIS Frontend

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

%I A265610 #21 Sep 08 2022 08:46:14
%S A265610 -1,0,2,11,49,204,825,3289,13013,51272,201552,791350,3105322,12183560,
%T A265610 47805615,187623765,736618125,2893125840,11367801060,44686512090,
%U A265610 175739405790,691437981000,2721606268290,10717182330426,42219554975874,166386610183024,655976895434000
%N A265610 a(n) = rf(n, n+2)/(n+2)! - rf(n, n)/n!, rf the rising factorial A265609.
%F A265610 G.f.: (-x^2+x-1-(x^2+3*x-1)/sqrt(1-4*x))/(2*x^2).
%F A265610 a(n) = binomial(2*n+1, n-1)-(0^n + binomial(2*n, n))/2 = A002054(n) - A088218(n).
%F A265610 a(n) = (1/2)*(3*n+2)*(n+2)*(n^2-1)*Gamma(2*n+1)/Gamma(n+3)^2 for n>=1.
%F A265610 a(n) ~ 4^n*(3/2)/sqrt(n*Pi).
%t A265610 Join[{-1}, Table[Binomial[2 n + 1, n - 1] - Binomial[2 n, n]/2, {n, 1, 36}]] (* _Vincenzo Librandi_, Dec 20 2015 *)
%o A265610 (Sage)
%o A265610 A265610 = lambda n: rising_factorial(n, n+2)/factorial(n+2) - rising_factorial(n, n)/factorial(n)
%o A265610 print([A265610(n) for n in srange(27)])
%o A265610 (Magma) [Binomial(2*n+1, n-1)-(0^n + Binomial(2*n, n))/2: n in [0..30]]; // _Vincenzo Librandi_, Dec 20 2015
%Y A265610 Cf. A002054, A088218.
%K A265610 sign
%O A265610 0,3
%A A265610 _Peter Luschny_, Dec 19 2015