A078820 - OEIS

%I #14 Feb 16 2023 05:07:43

%S 5,24,100,400,1575,6160,24024,93600,364650,1421200,5542680,21633248,

%T 84504875,330372000,1292646000,5061729600,19835652870,77786874000,

%U 305254551000,1198665468000,4709756401350,18516070880736,72834194898000,286645366072000,1128666128908500

%N a(n) = 20*C(2n,n)*(2n+1)/(n+4).

%F D-finite with recurrence a(n) = a(n-1)*(4n^2+14n+6)/(n^2+4n) = A078817(3, n) = (2n+1)*A078819(n)/7 = 20*A002457(n)/(n+4).

%F From _Amiram Eldar_, Feb 16 2023: (Start)

%F Sum_{n>=0} 1/a(n) = 11*Pi/(90*sqrt(3)) + 1/30.

%F Sum_{n>=0} (-1)^n/a(n) = 17*log(phi)/(25*sqrt(5)) + 1/50, where phi is the golden ratio (A001622). (End)

%e a(5)=20*C(10,5)*11/9=6160.

%t Table[20Binomial[2n,n] (2n+1)/(n+4),{n,0,30}] (* _Harvey P. Dale_, Nov 02 2011 *)

%Y Cf. A001622, A001700, A001791, A002457, A007946, A078817, A078819.

%K nonn,easy

%O 0,1

%A _Henry Bottomley_, Dec 07 2002