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%I #7 Aug 04 2017 01:14:31
%S 1,1,4,15,-14,185,-454,2107,-6194,22689,-70058,234971,-734304,2368379,
%T -7404318,23417955,-72988938,228324569,-708982738,2202742447,
%U -6815736144,21077285943,-65016664062,200371842727,-616463969324,1894794918275,-5816606133674,17839764136377
%N Row sums of the swinging derangement triangle (A163770).
%H G. C. Greubel, <a href="/A163773/b163773.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html"> Swinging Factorial</a>.
%F a(n) = Sum_{k=0..n} Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).
%p swing := proc(n) option remember; if n = 0 then 1 elif
%p irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p a := proc(n) local i,k; add(add((-1)^(n-i)*binomial(n-k,n-i)*swing(i),i=k..n), k=0..n) end:
%t sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Aug 03 2017 *)
%Y Cf. A163770.
%K sign
%O 0,3
%A _Peter Luschny_, Aug 05 2009
%E Terms a(18) onward added by _G. C. Greubel_, Aug 03 2017
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