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A231531 Imaginary part of Product_{k=1..n} (k+I). 8
0, 1, 3, 10, 40, 190, 1050, 6620, 46800, 365300, 3103100, 28269800, 271627200, 2691559000, 26495469000, 238131478000, 1394099824000, -15194495654000, -936096296850000, -29697351895900000, -819329864480400000, -21683886333440500000, -570263312237604700000, -15145164178973569000000, -409583160925827252000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 shows the imaginary parts of factim(n,1). The real parts are in A231530 and squares of magnitudes are in A101686.

LINKS

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

FORMULA

From Vladimir Reshetnikov, Oct 22 2015 : (Start)

a(n) = Im((1+i)_n) = -Re(Gamma(i)*Gamma(n+1-i))*sinh(Pi)/Pi, where (a)_n is the Pochhammer symbol, i=sqrt(-1).

a(n) = (-1)^n*A003703(n+1).

E.g.f.: sin(log(1-x))/(x-1). (End)

EXAMPLE

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

MAPLE

seq(simplify(-sinh(Pi)*Im(I!*(n-I)!)/Pi), n=0..19); # Peter Luschny, Oct 23 2015

MATHEMATICA

Table[Im[Pochhammer[1+I, n]], {n, 0, 20}]

Table[Sum[(-1)^(n+k) StirlingS1[n+1, 2k], {k, 0, (n+1)/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)

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); imag(a)

(PARI) t(n) = if( n<0, 0, n! * polcoeff(cos(log(1+x+x*O(x^n))), n));

vector(50, n, n--; (-1)^n*t(n+1)) \\ Altug Alkan, Oct 22 2015

CROSSREFS

Cf. A231530 (real parts), A101686 (squares of magnitudes), A003703.

See A242651, A242652 for a pair of similar sequences.

Sequence in context: A216367 A003703 A242651 * A136128 A089902 A093133

Adjacent sequences:  A231528 A231529 A231530 * A231532 A231533 A231534

KEYWORD

sign,easy

AUTHOR

Stanislav Sykora, Nov 10 2013

STATUS

approved

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Last modified May 29 21:17 EDT 2016. Contains 273510 sequences.