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

1, 1, 1, 0, -10, -90, -730, -6160, -55900, -549900, -5864300, -67610400, -839594600, -11186357000, -159300557000, -2416003824000, -38894192662000, -662595375078000, -11911522255750000, -225382826562400000, -4477959179352100000, -93217812901913700000, -2029107997508660900000

Offset: 0

Author

Stanislav Sykora, Nov 10 2013

Keywords

• sign
• easy

Comments

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.

Examples

			factim(5,1) = -90 + 190*i. Hence a(5) = -90.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A231531 (imaginary parts), A101686 (squares of magnitudes), A009454.

See A242651, A242652 for a pair of similar sequences.

Programs

Maple

seq(simplify(Re(I!*(n-I)!)*sinh(Pi)/Pi),n=0..22); # Peter Luschny, Oct 23 2015

Mathematica

Table[Re[Product[k+I,{k,n}]],{n,0,30}] (* Harvey P. Dale, Aug 04 2016 *)

PARI

Factim(nmax,m)={local(a,k);a=vector(nmax);a[1]=1+0*I;
  for (k=2,nmax,a[k]=a[k-1]*(k-1+m*I););return(a);}
a = Factim(1000,1); real(a)

Python

from sympy.functions.combinatorial.numbers import stirling
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

Sage

A231530 = lambda n : rising_factorial(1-I, n).real()
[A231530(n) for n in range(24)] # Peter Luschny, Oct 23 2015

Formula

From Peter Luschny, Oct 23 2015: (Start)

a(n) = Re(i!*(n-i)!)*sinh(Pi)/Pi.

a(n) = n!*[x^n](cos(log(1-x))/(1-x)).

a(n) = Sum_{k=0..floor(n/2)} (-1)^(n+k)*Stirling1(n+1,2*k+1).

a(n) = Re(rf(1+i,n)) where rf(k,n) is the rising factorial and i the imaginary unit.

a(n) = (-1)^n*A009454(n+1). (End)