A119575 - OEIS

%I #18 Oct 23 2023 12:28:00

%S 9,16,50,180,686,2688,10692,42900,173030,700128,2838524,11522056,

%T 46802700,190182400,772913160,3141129780,12764118870,51857916000,

%U 210638666700,855355383960,3472419702180,14092569803520,57176602275000,231908298827400,940340123399196,3811765978738368

%N a(n) = binomial(2*n,n)*(n+3)^2/(n+1).

%F From _Stefano Spezia_, Aug 24 2023: (Start)

%F O.g.f.: (2*(sqrt(1 - 4*x) - 1) + x*(21 - 8*sqrt(1 - 4*x) - 50*x))/(x*(1 - 4*x)^(3/2)).

%F E.g.f.: exp(2*x)*((9 + 2*x)*BesselI(0, 2*x) + 2*(x - 2)*BesselI(1, 2*x)).

%F a(n) ~ c*4^n*sqrt(n), where c = A087197. (End)

%p [seq (binomial(2*n,n)*(n+3)^2/(n+1),n=0..25)];

%t a[n_] := Binomial[2*n, n]*(n + 3)^2/(n + 1); Table[a[n], {n, 0, 25}] (* _Robert P. P. McKone_, Aug 25 2023 *)

%o (PARI) a(n) = binomial(2*n,n)/(n+1)*(n+3)^2 \\ _Charles R Greathouse IV_, Oct 23 2023

%Y Cf. A000984, A087197.

%K easy,nonn

%O 0,1

%A _Zerinvary Lajos_, May 31 2006

%E More terms from _Stefano Spezia_, Aug 24 2023