OFFSET
0,4
COMMENTS
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
FORMULA
a(n) = Sum_{k=0..floor(n/2)} ceiling(2^(k-2))*A106828(n, k).
a(n) ~ n * n! / (4*exp(2)). - Vaclav Kotesovec, Sep 10 2021
EXAMPLE
E.g.f.: 1 + x^2/2! + 2*x^3/3! + 9*x^4/4! + 44*x^5/5! + 280*x^6/6! + 2064*x^7/7! + 17528*x^8/8! + 167488*x^9/9! + ...
a(11) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(11, k) = 20721920.
For k = 0, A106828(11,0) = 0.
For k = 1, ceiling(2^(1-2))*A106828(11, 1) == -1 (mod 11), because ceiling(2^(1-2)) = 1 and A106828(11, 1) = (11-1)!
For k >= 2, ceiling(2^(k-2))*A106828(11, k) == 0 (mod 11), because A106828(11, k) == 0 (mod 11), result a(11) == -1 (mod 11).
a(10) = Sum_{k=0..5} ceiling(2^(k-2))*A106828(10, k) = 1777536.
a(10) == 0 (mod (10-1)), because for k >= 0, A106828(10, k) == 0 (mod 9).
MAPLE
a := series((-1-2*x-2*log(1-x)+exp(-2*x)/(1-x)^2)/4+1, x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
# second program:
a := n -> add(ceil(2^(k-2))*A106828(n, k), k=0..iquo(n, 2)):
seq(a(n), n=0..23);
MATHEMATICA
CoefficientList[Series[(-1 - 2*x - 2*Log[1 - x] + Exp[-2*x]/(1 - x)^2)/4 + 1, {x, 0, 23}], x]*Range[0, 23]!
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace((-1-2*x-2*log(1-x)+exp(-2*x)/(1-x)^2)/4 + 1)) \\ Michel Marcus, Sep 07 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Mélika Tebni, Sep 07 2021
STATUS
approved