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 A336634 #9 Jul 30 2020 18:25:51
%S A336634 1,1,0,-4,14,-18,-168,1920,-11898,27398,582896,-13028904,183020620,
%T A336634 -2061910004,17930433744,-65293856160,-1965585556410,69343044999750,
%U A336634 -1519055329884960,26755366818127560,-374375460816570780,2924763867241325220
%N A336634 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-x) * BesselI(0,2*sqrt(x))^2.
%H A336634 Robert Israel, <a href="/A336634/b336634.txt">Table of n, a(n) for n = 0..450</a>
%F A336634 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * binomial(2*k,k) * (n-k)!.
%F A336634 D-finite with recurrence: n*a(n) = -(3*n^2 - 7*n + 3)*a(n - 1) + (7 - 3*n)*(n - 1)^2*a(n - 2) - (n - 1)^2*(n - 2)^2*a(n - 3). - _Robert Israel_, Jul 30 2020
%p A336634 rec:= n*a(n) = -(3*n^2 - 7*n + 3)*a(n - 1) + (7 - 3*n)*(n - 1)^2*a(n - 2) - (n - 1)^2*(n - 2)^2*a(n - 3):
%p A336634 f:= gfun:-rectoproc({rec,a(0)=1,a(1)=1,a(2)=0},a(n),remember):
%p A336634 map(f, [$0..30]); # _Robert Israel_, Jul 30 2020
%t A336634 nmax = 21; CoefficientList[Series[Exp[-x] BesselI[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2
%t A336634 Table[(-1)^n n! HypergeometricPFQ[{1/2, -n}, {1, 1}, 4], {n, 0, 21}]
%t A336634 Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[2 k, k] (n - k)!, {k, 0, n}], {n, 0, 21}]
%o A336634 (PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k)^2 * binomial(2*k,k) * (n-k)!); \\ _Michel Marcus_, Jul 30 2020
%Y A336634 Cf. A000984, A009940, A244973, A336293.
%K A336634 sign
%O A336634 0,4
%A A336634 _Ilya Gutkovskiy_, Jul 28 2020