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 A231531 Imaginary part of Product_{k=1..n} (k+I). 9
 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 October 23 14:54 EDT 2019. Contains 328345 sequences. (Running on oeis4.)