This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378803 #7 Dec 08 2024 02:35:42
%S A378803 0,4,112,1980,29120,387600,4845456,58017960,673171200,7625605680,
%T A378803 84766052800,927990034972,10032268963392,107317291572400,
%U A378803 1137727464904800,11968670068362000,125062895892372480,1299098807032012272,13423997084049034560,138068403550647828400,1414126456884869728000
%N A378803 a(n) = n^2 * binomial(4*n, n).
%H A378803 Amiram Eldar, <a href="/A378803/b378803.txt">Table of n, a(n) for n = 0..500</a>
%H A378803 Necdet Batir and Anthony Sofo, <a href="http://dx.doi.org/10.1016/j.amc.2013.05.053">On some series involving reciprocals of binomial coefficients</a>, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.
%F A378803 a(n) = n^2 * A005810(n).
%F A378803 a(n) = n * A378802(n).
%F A378803 a(n) == 0 (mod 4).
%F A378803 Sum_{n>=1} 1/a(n) = -(3/2)*log((c-1)/(c+1))^2 + (3/4) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1))^2 + (3/4) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1))^2 = 0.25947076781691783..., where c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013, p. 336, Example 3).
%F A378803 Sum_{n>=1} (-1)^n/a(n) = -(3/2)*log((1-d)/(1+d))^2 + (3/4) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1))^2 + (3/4) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1))^2 = -0.24154452788843591937..., where d = sqrt(1 - (8/sqrt(3))*(((3*sqrt(3)+sqrt(283))/16)^(1/3) - (((3*sqrt(3)+sqrt(283))/16)^(-1/3)))) (Batir and Sofo, 2013, pp. 336-337, Example 4).
%t A378803 a[n_] := n^2 * Binomial[4*n, n]; Array[a, 20, 0]
%o A378803 (PARI) a(n) = n^2 * binomial(4*n, n);
%Y A378803 Cf. A005810, A378802.
%K A378803 nonn,easy
%O A378803 0,2
%A A378803 _Amiram Eldar_, Dec 07 2024