A119575 a(n) = binomial(2*n,n)*(n+3)^2/(n+1).
9, 16, 50, 180, 686, 2688, 10692, 42900, 173030, 700128, 2838524, 11522056, 46802700, 190182400, 772913160, 3141129780, 12764118870, 51857916000, 210638666700, 855355383960, 3472419702180, 14092569803520, 57176602275000, 231908298827400, 940340123399196, 3811765978738368
Offset: 0
Programs
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Maple
[seq (binomial(2*n,n)*(n+3)^2/(n+1),n=0..25)];
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Mathematica
a[n_] := Binomial[2*n, n]*(n + 3)^2/(n + 1); Table[a[n], {n, 0, 25}] (* Robert P. P. McKone, Aug 25 2023 *) -
PARI
a(n) = binomial(2*n,n)/(n+1)*(n+3)^2 \\ Charles R Greathouse IV, Oct 23 2023
Formula
From Stefano Spezia, Aug 24 2023: (Start)
O.g.f.: (2*(sqrt(1 - 4*x) - 1) + x*(21 - 8*sqrt(1 - 4*x) - 50*x))/(x*(1 - 4*x)^(3/2)).
E.g.f.: exp(2*x)*((9 + 2*x)*BesselI(0, 2*x) + 2*(x - 2)*BesselI(1, 2*x)).
a(n) ~ c*4^n*sqrt(n), where c = A087197. (End)
Extensions
More terms from Stefano Spezia, Aug 24 2023