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 A180364 #28 Sep 02 2014 07:14:00
%S A180364 1,19,205,1839,14961,114483,839917,5975455,41524897,283272723,
%T A180364 1903686093,12636115407,83007985425,540484102707,3492471392493,
%U A180364 22418010385983,143062290575937,908253002030355,5739641232682957,36121371405797743,226475167518421681
%N A180364 a(n) = sum_{k=0..n} C(n,k)*C(n+k,k)*(2*k+1)^2, where C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).
%C A180364 For any n > 0, we have a(0)+ ... + a(n-1) = n*sum_{k=0}^{n-1} (2*k+1)*C(n-1,k)*C(n+k,k) = n^2*A002002(n). The first equality can be easily deduced, and the second equality holds since both sides satisfy the same recurrence by the Zeilberger algorithm.
%C A180364 Conjecture: The sequence a(n+1)/a(n) (n = 0,1,...) is strictly decreasing to the limit 3+2*sqrt(2), and the sequence a(n+1)^(1/(n+1))/a(n)^(1/n) (n = 1,2,3,..) is strictly increasing to the limit 1.
%H A180364 Zhi-Wei Sun, <a href="/A180364/b180364.txt">Table of n, a(n) for n = 0..200</a>
%H A180364 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1408.5381">Two kinds of numbers and their arithmetic properties</a>, arXiv:1408.5381, 2014.
%F A180364 Recurrence (obtained via the Zeilberger algorithm):
%F A180364 -(n+1)*(2*n^2+10*n+11)*a(n) + (2*n+3)*(6*n^2+18*n-7)*a(n+1) - (n+2)*(2*n^2+2*n-1)*a(n+2) = 0.
%F A180364 a(n) ~ sqrt(8+6*sqrt(2)) * (3+2*sqrt(2))^n * n^(3/2) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Sep 02 2014
%e A180364 a(1) = 19 since sum_{k=0,1} C(1,k)*C(1+k,k)*(2k+1)^2 = 1 + 2*3^2 = 19.
%t A180364 a[n_] := Sum[Binomial[n, k] Binomial[n + k, k] (2 k + 1)^2, {k, 0, n}]
%t A180364 Table[a[n], {n, 0, 20}]
%Y A180364 Cf. A002002, A246460, A246542, A246543.
%K A180364 nonn
%O A180364 0,2
%A A180364 _Zhi-Wei Sun_, Aug 29 2014