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%I #10 May 08 2020 17:34:18
%S 1,11,73,403,2021,9567,43611,193683,844213,3629083,15437951,65143503,
%T 273148279,1139548469,4734740493,19606960755,80969809797,333601494651
%N Row sums of triangle A163772.
%H G. C. Greubel, <a href="/A163775/b163775.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%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)*(2i+1)$ 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(2*i+1),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[2*i + 1], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Aug 04 2017 *)
%Y Cf. A163772.
%K nonn
%O 0,2
%A _Peter Luschny_, Aug 05 2009
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