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A105750 Real part of Product_{k = 0..n} (1 + k*i), i = sqrt(-1).

%I A105750 #44 Feb 22 2024 17:46:05
%S A105750 1,1,-1,-10,-10,190,730,-6620,-55900,365300,5864300,-28269800,
%T A105750 -839594600,2691559000,159300557000,-238131478000,-38894192662000,
%U A105750 -15194495654000,11911522255750000,29697351895900000,-4477959179352100000,-21683886333440500000,2029107997508660900000
%N A105750 Real part of Product_{k = 0..n} (1 + k*i), i = sqrt(-1).
%C A105750 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
%C A105750 From _Peter Bala_, Jun 03 2023: (Start)
%C A105750 Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes.
%C A105750 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}.
%C A105750 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}.
%C A105750 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.
%C A105750 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).
%C A105750 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)
%H A105750 N. J. A. Sloane, <a href="/A105750/b105750.txt">Table of n, a(n) for n = 0..100</a>
%H A105750 V. H. Moll, <a href="http://www.tulane.edu/~vhm/papers_html/xn-final.pdf">An arithmetic conjecture on a sequence of arctangent sums</a>, 2012, see f_n.
%F A105750 a(n) = Re( Product_{k = 0..n} (1 - k*i) ).
%F A105750 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
%F A105750 From _Peter Bala_, May 28 2023: (Start)
%F A105750 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).
%F A105750 (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.
%F A105750 a(2*n) = (-1)^n*A009454(2*n+1) for n >= 0.
%F A105750 a(2*n-1) = (-1)^n*A003703(2*n) for n >= 1. (End)
%p A105750 A105750 := proc(n)
%p A105750     mul(1-k*I,k=0..n) ;
%p A105750     Re(%) ;
%p A105750 end proc: # _R. J. Mathar_, Jan 04 2013
%t A105750 x[n_] := x[n] = If[n == 1, 1, (x[n-1]+n)/(1-n*x[n-1])];
%t A105750 u[n_] := n*x[n-1]-1;
%t A105750 f[n_] := f[n] = If[n == 1, 1, u[n]*f[n-1]];
%t A105750 a[n_] := If[n == 0, 1, (-1)^(n+1)*f[n]];
%t A105750 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 17 2023, after _N. J. A. Sloane_ *)
%o A105750 (Python)
%o A105750 from sympy.functions.combinatorial.numbers import stirling
%o A105750 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
%Y A105750 Cf. A003703, A009454, A105751, A220448, A220449, A363409 - A363416.
%K A105750 easy,sign
%O A105750 0,4
%A A105750 _Paul Barry_, Apr 18 2005
%E A105750 Corrected by _N. J. A. Sloane_, Nov 05 2005