|
%I #13 Jun 19 2022 08:29:24
%S 1,0,-5,22,9,-1244,14335,-79470,-586943,25131304,-434574909,
%T 4418399470,8524321465,-1771817986548,53502570125719,
%U -1052208254769014,11804172888840705,131741085049224400,-12970386000411511733,482732550618027365574,-12599999790172579025879
%N a(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(n, k)*|A021009(n, k)|.
%C Related to the coefficient triangle of generalized Laguerre polynomials A021009.
%F 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
%p T := proc(n, k) local S; S := proc(n, k) option remember;
%p `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 a := n -> add((-1)^(n-j)*T(n, j)*binomial(n, j), j=0..n): seq(a(n), n=0..20);
%o (PARI) rowT(n) = Vecrev(n!*pollaguerre(n)); \\ A021009
%o 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 Cf. A021009, A216831.
%K sign
%O 0,3
%A _Peter Luschny_, May 04 2021
|
