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A231530 Real part of Product_{k=1..n} (k+I). 9
1, 1, 1, 0, -10, -90, -730, -6160, -55900, -549900, -5864300, -67610400, -839594600, -11186357000, -159300557000, -2416003824000, -38894192662000, -662595375078000, -11911522255750000, -225382826562400000, -4477959179352100000, -93217812901913700000, -2029107997508660900000 (list; graph; refs; listen; history; text; internal format)
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). 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

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

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(n) = -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

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 A162756 A199940

Adjacent sequences:  A231527 A231528 A231529 * A231531 A231532 A231533

KEYWORD

sign,easy

AUTHOR

Stanislav Sykora, Nov 10 2013

STATUS

approved

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Last modified December 8 17:11 EST 2016. Contains 278946 sequences.