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A231530 Real part of Product_{k=1..n} (k+i), where i is the imaginary unit.

%I A231530 #32 Feb 22 2024 12:15:46
%S A231530 1,1,1,0,-10,-90,-730,-6160,-55900,-549900,-5864300,-67610400,
%T A231530 -839594600,-11186357000,-159300557000,-2416003824000,-38894192662000,
%U A231530 -662595375078000,-11911522255750000,-225382826562400000,-4477959179352100000,-93217812901913700000,-2029107997508660900000
%N A231530 Real part of Product_{k=1..n} (k+i), where i is the imaginary unit.
%C A231530 Extension of factorial(n) to factim(n,m) defined by the recurrence a(0)=1, a(n) = a(n-1)*(n+m*i), where i is the imaginary unit. Hence n! = factim(n,0), while the current sequence lists the real parts of factim(n,1). The imaginary parts are in A231531 and squares of magnitudes are in A101686.
%H A231530 Stanislav Sykora, <a href="/A231530/b231530.txt">Table of n, a(n) for n = 0..440</a>
%F A231530 From _Peter Luschny_, Oct 23 2015: (Start)
%F A231530 a(n) = Re(i!*(n-i)!)*sinh(Pi)/Pi.
%F A231530 a(n) = n!*[x^n](cos(log(1-x))/(1-x)).
%F A231530 a(n) = Sum_{k=0..floor(n/2)} (-1)^(n+k)*Stirling1(n+1,2*k+1).
%F A231530 a(n) = Re(rf(1+i,n)) where rf(k,n) is the rising factorial and i the imaginary unit.
%F A231530 a(n) = (-1)^n*A009454(n+1). (End)
%e A231530 factim(5,1) = -90 + 190*i. Hence a(5) = -90.
%p A231530 seq(simplify(Re(I!*(n-I)!)*sinh(Pi)/Pi),n=0..22); # _Peter Luschny_, Oct 23 2015
%t A231530 Table[Re[Product[k+I,{k,n}]],{n,0,30}] (* _Harvey P. Dale_, Aug 04 2016 *)
%o A231530 (PARI) Factim(nmax,m)={local(a,k);a=vector(nmax);a[1]=1+0*I;
%o A231530   for (k=2,nmax,a[k]=a[k-1]*(k-1+m*I););return(a);}
%o A231530 a = Factim(1000,1); real(a)
%o A231530 (Sage)
%o A231530 A231530 = lambda n : rising_factorial(1-I, n).real()
%o A231530 [A231530(n) for n in range(24)] # _Peter Luschny_, Oct 23 2015
%o A231530 (Python)
%o A231530 from sympy.functions.combinatorial.numbers import stirling
%o A231530 def A231530(n): return sum(stirling(n+1,(k<<1)+1,kind=1)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # _Chai Wah Wu_, Feb 22 2024
%Y A231530 Cf. A231531 (imaginary parts), A101686 (squares of magnitudes), A009454.
%Y A231530 See A242651, A242652 for a pair of similar sequences.
%K A231530 sign,easy
%O A231530 0,5
%A A231530 _Stanislav Sykora_, Nov 10 2013