OFFSET
0,4
FORMULA
E.g.f. y(x) satisfies y*y' = exp(-x)*x/(1-x)^2.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
a(n) ~ 2 * n^n / exp(n + 1/2). - Vaclav Kotesovec, Jul 06 2021
EXAMPLE
sqrt(-1+2/(1-x)/exp(x)) = 1 + x^2/2! + 2*x^3/3! + 6*x^4/4! + 24*x^5/5! + 135*x^6/6! + 930*x^7/7! + 7105*x^8/8! + 59192*x^9/9! + ...
For k=1, (-1)^(1-1)*A014304(1-1)*A008306(23,1) == -1 (mod 23), because A014304(0) = 1 and A008306(23,1) = (23-1)!
For k>=2, (-1)^(k-1)*A014304(k-1)*A008306(23,k) == 0 (mod 23), because A008306(23,k) == 0 (mod 23), result a(23) == -1 (mod 23).
a(18) == 0 (mod (18-1)), because for k >= 1, A008306(18,k) == 0 (mod 17).
MAPLE
A014304:= proc(n) option remember; `if`(n=0, 1, (-1)^n + add(binomial(n, k)*A014304(k)* A014304(n-k-1), k=0..n-1)) end:
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
# second program:
a := series(sqrt(-1+2/(1-x)/exp(x)), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
MATHEMATICA
CoefficientList[Series[Sqrt[-1+2/(1-x)/E^x], {x, 0, 24}], x] * Range[0, 24]!
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace(sqrt(-1 + 2 / (1 - x) / exp(x)))) \\ Michel Marcus, Jul 06 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Mélika Tebni, Jul 06 2021
STATUS
approved