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A231530
Real part of Product_{k=1..n} (k+i), where i is the imaginary unit.
10
1, 1, 1, 0, -10, -90, -730, -6160, -55900, -549900, -5864300, -67610400, -839594600, -11186357000, -159300557000, -2416003824000, -38894192662000, -662595375078000, -11911522255750000, -225382826562400000, -4477959179352100000, -93217812901913700000, -2029107997508660900000
OFFSET
0,5
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.
LINKS
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)
EXAMPLE
factim(5,1) = -90 + 190*i. Hence a(5) = -90.
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 *)
PROG
(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)
(Sage)
A231530 = lambda n : rising_factorial(1-I, n).real()
[A231530(n) for n in range(24)] # Peter Luschny, Oct 23 2015
(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
CROSSREFS
Cf. A231531 (imaginary parts), A101686 (squares of magnitudes), A009454.
See A242651, A242652 for a pair of similar sequences.
Sequence in context: A265325 A038726 A009454 * A242652 A291392 A162756
KEYWORD
sign,easy
AUTHOR
Stanislav Sykora, Nov 10 2013
STATUS
approved