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%I #9 Aug 06 2017 22:43:50
%S 1,3,10,41,116,427,1240,4181,12472,40091,121364,380701,1160186,
%T 3593969,10979532,33785469,103258800,316532947,966976444,2957131673,
%U 9026437602,27558146133,84043120308,256263107177,780817641926
%N Row sums of triangle A163840.
%H G. C. Greubel, <a href="/A163843/b163843.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} 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(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[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 06 2017 *)
%Y Cf. A056040, A163840.
%K nonn
%O 0,2
%A _Peter Luschny_, Aug 06 2009
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