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 A073701 #29 Jul 15 2025 18:06:58
%S A073701 1,0,1,8,129,3224,116065,5687184,363979777,29482361936,2948236193601,
%T A073701 356736579425720,51370067437303681,8681541396904322088,
%U A073701 1701582113793247129249,382855975603480604081024,98011129754491034644742145,28325216499047909012330479904,9177370145691522519995075488897
%N A073701 a(n) = n^2*a(n-1)+(-1)^n.
%C A073701 The sequence b(n) := n!^2 satisfies the same recurrence below for a(n) with the initial conditions b(0) = 1, b(1) = 1. It follows that, for n >=3, a(n) = n!^2*(1/(4 + 4/(8 + 9/(15 +...+ (n-1)^2/(n^2-1))))). Hence BesselJ(0,2) := sum {k = 0..inf} (-1)^k/k!^2 = 1/(4 + 4/(8 + 9/(15 + ...+(n-1)^2/(n^2+1 + ...)))) = 0.22388 90779 ... . Cf. A006040. - _Peter Bala_, Jul 09 2008
%H A073701 Seiichi Manyama, <a href="/A073701/b073701.txt">Table of n, a(n) for n = 0..253</a>
%F A073701 a(n) = n!^2*Sum_{k=0..n} (-1)^k/k!^2.
%F A073701 BesselJ(0, 2*sqrt(x))/(1-x) = Sum_{n>=0} a(n)*x^n/n!^2.
%F A073701 a(n) = round(n!^2*BesselJ(0, 2)), n>0.
%F A073701 Recurrence: a(0) = 1, a(1) = 0, a(n) = (n^2-1)*a(n-1) + (n-1)^2*a(n-2), n >= 2. - _Peter Bala_, Jul 09 2008, corrected by _Georg Fischer_, Feb 13 2020
%F A073701 a(n) = Sum_{k=0..n} (-1)^(n-k)*(k!*binomial(n,k))^2. - _Ridouane Oudra_, Jul 11 2025
%t A073701 Join[{a = 1}, Table[a = a*n^2 + (-1)^n, {n, 15}]] (* _Jayanta Basu_, Jul 08 2013 *)
%o A073701 (Magma) [1] cat [ n eq 1 select 0 else n^2*Self(n-1)+(-1)^n:n in [1..15]]; // _Marius A. Burtea_, Feb 13 2020
%Y A073701 Cf. A000166, A006040.
%K A073701 nonn
%O A073701 0,4
%A A073701 _Vladeta Jovovic_, Aug 30 2002