login
a(n) = 6 * binomial(2*n,n-1) + binomial(2*n-1,n).
2

%I #20 Apr 03 2024 05:02:33

%S 7,27,100,371,1386,5214,19734,75075,286858,1100138,4232592,16328942,

%T 63146500,244711260,950094810,3694876515,14390571690,56122547250,

%U 219140635560,856617714810,3351878581740,13127747882340,51458942047500,201869999056206,792497263436676

%N a(n) = 6 * binomial(2*n,n-1) + binomial(2*n-1,n).

%H Harvey P. Dale, <a href="/A185080/b185080.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A046902(2*n,n) (Central terms of Clark's triangle).

%F a(n) = 6 * A007318(2*n,n-1) + A007318(2*n-1,n).

%F From _G. C. Greubel_, Apr 03 2024: (Start)

%F a(n) = (13*n+1)*A000108(n)/2.

%F a(n) = (2 + 22*n - 52*n^2)*a(n-1)/(12 - n - 13*n^2).

%F G.f.: ((6 - 11*x)*sqrt(1-4*x) - (1-4*x)*(6+x))/(2*x*(1-4*x)).

%F E.g.f.: (1/2)*(-1 + exp(2*x)*(BesselI(0, 2*x) + 12*BesselI(1, 2*x))).(End)

%t Table[6Binomial[2n,n-1]+Binomial[2n-1,n],{n,30}] (* _Harvey P. Dale_, Dec 28 2012 *)

%o (Haskell)

%o a185080 n = 6 * a007318 (2 * n) (n - 1) + a007318 (2 * n - 1) n

%o (Magma) [(13*n+1)*Catalan(n)/2: n in [1..40]]; // _G. C. Greubel_, Apr 03 2024

%o (SageMath) [(13*n+1)*binomial(2*n,n)/(2*n+2) for n in range(1,41)] # _G. C. Greubel_, Apr 03 2024

%Y Cf. A000108, A007318, A024482, A046224, A046902.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Dec 26 2012