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Real part of Product_{k=0..n} (i-k), where i = sqrt(-1).
3

%I #29 May 23 2021 02:58:46

%S 0,-1,3,-10,40,-190,1050,-6620,46800,-365300,3103100,-28269800,

%T 271627200,-2691559000,26495469000,-238131478000,1394099824000,

%U 15194495654000,-936096296850000,29697351895900000,-819329864480400000,21683886333440500000,-570263312237604700000,15145164178973569000000

%N Real part of Product_{k=0..n} (i-k), where i = sqrt(-1).

%C Shifted version of A003703. - _R. J. Mathar_, May 30 2014

%D Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 561-562, Section 148.

%F a(n) = Sum_{k=0..floor((n+1)/2)} (-1)^k*Stirling1(n+1,2*k). - _Ammar Khatab_, May 23 2021

%e Table of n, Product_{k=0..n} (i-k):

%e 0, i

%e 1, -1 - i

%e 2, 3 + i

%e 3, -10

%e 4, 40 - 10*i

%e 5, -190 + 90*i

%e 6, 1050 - 730*i

%e 7, -6620 + 6160*i

%e 8, 46800 - 55900*i

%e 9, -365300 + 549900*i

%e 10, 3103100 - 5864300*i

%e 11, -28269800 + 67610400*i

%e 12, 271627200 - 839594600*i

%t Table[Re[(I - n)*Pochhammer[1 + I - n, n]], {n, 0, 25}] (* _Vaclav Kotesovec_, May 23 2021 *)

%o (PARI) a(n) = real(prod(k=0, n, I-k)); \\ _Michel Marcus_, Jan 03 2021

%Y Cf. A003703, A242652, A101686.

%Y A231531 is the same except for signs.

%K sign

%O 0,3

%A _N. J. A. Sloane_, May 29 2014