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 A343840 #18 Jul 11 2025 15:15:45
%S A343840 1,0,-5,22,9,-1244,14335,-79470,-586943,25131304,-434574909,
%T A343840 4418399470,8524321465,-1771817986548,53502570125719,
%U A343840 -1052208254769014,11804172888840705,131741085049224400,-12970386000411511733,482732550618027365574,-12599999790172579025879
%N A343840 a(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(n, k)*|A021009(n, k)|.
%C A343840 Related to the coefficient triangle of generalized Laguerre polynomials A021009.
%F A343840 Sum_{n>=0} a(n) * x^n / n!^3 = BesselJ(0,2*sqrt(x)) * Sum_{n>=0} x^n / n!^3. - _Ilya Gutkovskiy_, Jun 19 2022
%F A343840 a(n) = Sum_{k=0..n} (-1)^k*k!*binomial(n,k)^3. - _Ridouane Oudra_, Jul 11 2025
%F A343840 Recurrence: n*(8*n - 11)*a(n) = -(n-1)*(24*n^2 - 49*n + 21)*a(n-1) - (n-1)*(24*n^3 - 33*n^2 - 14*n + 18)*a(n-2) - (n-2)^3*(n-1)*(8*n - 3)*a(n-3). - _Vaclav Kotesovec_, Jul 11 2025
%p A343840 T := proc(n, k) local S; S := proc(n, k) option remember;
%p A343840 `if`(k = 0, 1, `if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
%p A343840 a := n -> add((-1)^(n-j)*T(n, j)*binomial(n, j), j=0..n): seq(a(n), n=0..20);
%o A343840 (PARI) rowT(n) = Vecrev(n!*pollaguerre(n)); \\ A021009
%o A343840 a(n) = my(v=rowT(n)); sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(v[k+1])); \\ _Michel Marcus_, May 04 2021
%Y A343840 Cf. A021009, A216831.
%K A343840 sign
%O A343840 0,3
%A A343840 _Peter Luschny_, May 04 2021