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 A295113 #12 Jan 29 2018 02:59:26
%S A295113 9,13,17,219,1561,8169,39321,191389,985201,5430789,31943961,198471183,
%T A295113 1288665177,8665236121,59922226809,423935337411,3056528058577,
%U A295113 22392246851973,166311049602681,1250027314777795,9494339129623329,72784922204637153,562626619763553217,4381665416129531961,34354964747652467697
%N A295113 a(n) = (1/n)*Sum_{k=0..n-1} (8*k + 9)*A295112(k)^2.
%C A295113 Conjecture: a(n) is always integral for every n = 1,2,3,.... Moreover, for any odd prime p we have a(p) == 24 + 10*Leg(-1,p) - 18*Leg(3,p) - 9*Leg(p,3) (mod p^2), where Leg(m,p) denotes the Legendre symbol (m/p).
%C A295113 We also observe that Sum_{k=0}^{p-1}A295112(k)^2 == 2 (mod p) for any prime p > 3.
%H A295113 Zhi-Wei Sun, <a href="/A295113/b295113.txt">Table of n, a(n) for n = 1..200</a>
%H A295113 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1801.08905">On Motzkin numbers and central trinomial coefficients</a>, arXiv:1801.08905 [math.CO], 2018. (See Conjecture 5.1.)
%e A295113 a(2) = 13 since (1/2)*Sum_{k=0..1} (8k + 9)*A295112(k)^2 = (1/2)*((8*0 + 9)*A295112(0)^2 + (8 + 9)*A295112(1)^2) = (1/2)*(9*(-1)^2 + 17*(-1)^2) = 13.
%t A295113 W[n_]:=W[n]=Sum[Binomial[n,2k]Binomial[2k,k]/(2k-1),{k,0,n/2}];
%t A295113 a[n_]:=a[n]=1/n*Sum[(8k+9)W[k]^2,{k,0,n-1}];
%t A295113 Table[a[n],{n,1,25}]
%Y A295113 Cf. A001006, A295112.
%K A295113 nonn
%O A295113 1,1
%A A295113 _Zhi-Wei Sun_, Nov 14 2017