OFFSET
0,4
COMMENTS
Define u(n) as in A220448 and set f(n) = u(n)*f(n-1) for n >= 2, with f(1)=1 (this defines A220449). Then a(0)=1; a(n) = (-1)^(n+1)*f(n) for n >= 1. - N. J. A. Sloane, Dec 22 2012
From Peter Bala, Jun 03 2023: (Start)
Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes.
Type 1: primes p that do not divide any element of the sequence {a(n)}. The first few type 1 primes appear to be {3, 7, 11, 23, 31, 47, 59}.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. The first few type 2 primes appear to be {2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97}.
We conjecture that the set of type 2 primes consists of primes p == 1 (mod 4), equivalently, rational primes that split in the field extension Q(sqrt(-1)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002144.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined).
We conjecture that the sets of type 1 and type 3 primes taken together consist of primes p == 3 (mod 4), equivalently, rational primes that remain inert in the field extension Q(sqrt(-1)) of Q. See A002145. (End)
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..100
V. H. Moll, An arithmetic conjecture on a sequence of arctangent sums, 2012, see f_n.
FORMULA
a(n) = Re( Product_{k = 0..n} (1 - k*i) ).
Conjecture: a(n) -3*a(n-1) +(n^2-n+3)*a(n-2) +(-n^2+4*n-5)*a(n-3)=0. - R. J. Mathar, May 23 2014
From Peter Bala, May 28 2023: (Start)
a(n) = Sum_{k = 0..floor((n + 1)/2)} (-1)^k*|Stirling1(n+1, n-2*k+1)|, where Stirling1(n, k) = A048994(n,k).
(n - 1)*a(n) = (2*n - 1)*a(n-1) - n*(n^2 - 2*n + 2)*a(n-2) with a(0) = a(1) = 1 (see Moll, equation 1.16). Mathar's third-order recurrence above follows easily from this.
a(2*n) = (-1)^n*A009454(2*n+1) for n >= 0.
a(2*n-1) = (-1)^n*A003703(2*n) for n >= 1. (End)
MAPLE
MATHEMATICA
x[n_] := x[n] = If[n == 1, 1, (x[n-1]+n)/(1-n*x[n-1])];
u[n_] := n*x[n-1]-1;
f[n_] := f[n] = If[n == 1, 1, u[n]*f[n-1]];
a[n_] := If[n == 0, 1, (-1)^(n+1)*f[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 17 2023, after N. J. A. Sloane *)
PROG
(Python)
from sympy.functions.combinatorial.numbers import stirling
def A105750(n): return sum(stirling(n+1, n+1-(k<<1), kind=1)*(-1 if k&1 else 1) for k in range((n+1>>1)+1)) # Chai Wah Wu, Feb 22 2024
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Apr 18 2005
EXTENSIONS
Corrected by N. J. A. Sloane, Nov 05 2005
STATUS
approved