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 A307844 #36 May 04 2019 16:32:17
%S A307844 1,1,9,55,1729,19251,1050841,16977129,1322929665,28017221059,
%T A307844 2839212609001,74390784295653,9283240524317761,289865990675075725,
%U A307844 42976734096778661817,1557837326400792009751,267561369300137776050177,11042876765198762014337235
%N A307844 Constant term in the expansion of (n/x + 1 + n*x)^n.
%C A307844 Also coefficient of x^n in the expansion of (1 + x + (n*x)^2)^n.
%C A307844 Also coefficient of x^n in the expansion of 1/sqrt(1 - 2*x + (1-4*n^2)*x^2).
%H A307844 Seiichi Manyama, <a href="/A307844/b307844.txt">Table of n, a(n) for n = 0..351</a>
%F A307844 a(n) = Sum_{k=0..n} (1-2*n)^(n-k) * n^k * binomial(n,k) * binomial(2*k,k).
%F A307844 a(n) = Sum_{k=0..n} (1+2*n)^(n-k) * (-n)^k * binomial(n,k) * binomial(2*k,k).
%F A307844 a(n) = Sum_{k=0..floor(n/2)} n^(2*k) *binomial(n,2*k) * binomial(2*k,k).
%F A307844 a(n) ~ (exp(1/2) + (-1)^n * exp(-1/2)) * 2^(n - 1/2) * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, May 02 2019
%t A307844 Flatten[{1, Table[Sum[(-1)^k * (2*n + 1)^(n-k) * n^k * Binomial[n,k] * Binomial[2*k,k], {k, 0, n}], {n, 1, 20}]}] (* _Vaclav Kotesovec_, May 02 2019 *)
%o A307844 (PARI) {a(n) = polcoef((n/x+1+n*x)^n, 0)}
%o A307844 (PARI) {a(n) = polcoef((1+x+(n*x)^2)^n, n)}
%o A307844 (PARI) {a(n) = sum(k=0, n, (1-2*n)^(n-k)*n^k*binomial(n, k)*binomial(2*k, k))}
%o A307844 (PARI) {a(n) = sum(k=0, n, (1+2*n)^(n-k)*(-n)^k*binomial(n, k)*binomial(2*k, k))}
%o A307844 (PARI) {a(n) = sum(k=0, n\2, n^(2*k)*binomial(n, 2*k)*binomial(2*k, k))}
%Y A307844 Main diagonal of A307847.
%Y A307844 Cf. A002426, A092366, A186925, A187018.
%K A307844 nonn
%O A307844 0,3
%A A307844 _Seiichi Manyama_, May 01 2019