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A231530
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Real part of Product_{k=1..n} (k+i), where i is the imaginary unit.
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10
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1, 1, 1, 0, -10, -90, -730, -6160, -55900, -549900, -5864300, -67610400, -839594600, -11186357000, -159300557000, -2416003824000, -38894192662000, -662595375078000, -11911522255750000, -225382826562400000, -4477959179352100000, -93217812901913700000, -2029107997508660900000
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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factim(5,1) = -90 + 190*i. Hence a(5) = -90.
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MAPLE
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seq(simplify(Re(I!*(n-I)!)*sinh(Pi)/Pi), n=0..22); # Peter Luschny, Oct 23 2015
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MATHEMATICA
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Table[Re[Product[k+I, {k, n}]], {n, 0, 30}] (* Harvey P. Dale, Aug 04 2016 *)
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PROG
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(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()
(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
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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