A343840 a(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(n, k)*|A021009(n, k)|.

1, 0, -5, 22, 9, -1244, 14335, -79470, -586943, 25131304, -434574909, 4418399470, 8524321465, -1771817986548, 53502570125719, -1052208254769014, 11804172888840705, 131741085049224400, -12970386000411511733, 482732550618027365574, -12599999790172579025879

Offset

0,3

Comments

Related to the coefficient triangle of generalized Laguerre polynomials A021009.

Links

Table of n, a(n) for n=0..20.

Formula

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

Maple

T := proc(n, k) local S; S := proc(n, k) option remember;
`if`(k = 0, 1, `if`(k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: n!*S(n, k) end:
a := n -> add((-1)^(n-j)*T(n, j)*binomial(n, j), j=0..n): seq(a(n), n=0..20);

Prog

(PARI) rowT(n) = Vecrev(n!*pollaguerre(n)); \\ A021009
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

Crossrefs

Cf. A021009, A216831.

Keyword

sign

Author

Peter Luschny, May 04 2021

Status

approved