This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A163844 #13 Aug 22 2025 04:09:08
%S A163844 1,5,25,125,621,3065,15051,73645,359485,1752125,8532591,41537105,
%T A163844 202200415,984526275,4795673085,23372376525,113978687085,556205251325,
%U A163844 2716129289775,13273197773125,64909884686595,317652752793975,1555587408645225,7623031579626625
%N A163844 Row sums of triangle A163841.
%H A163844 G. C. Greubel, <a href="/A163844/b163844.txt">Table of n, a(n) for n = 0..1000</a>
%H A163844 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163844 a(n) = Sum_{k=0..n} Sum_{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040).
%p A163844 swing := proc(n) option remember; if n = 0 then 1 elif
%p A163844 irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p A163844 a := proc(n) local i,k; add(add(binomial(n-k,n-i)*swing(2*i),i=k..n),k=0..n) end:
%t A163844 sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[Sum[t[n, k], {k, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Aug 06 2017 *)
%Y A163844 Cf. A056040, A163841.
%K A163844 nonn,changed
%O A163844 0,2
%A A163844 _Peter Luschny_, Aug 06 2009