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 A163845 #13 Aug 22 2025 04:30:21
%S A163845 1,13,109,765,4881,29369,169919,956237,5272945,28632525,153638211,
%T A163845 816715073,4309138419,22598433555,117926579385,612863125965,
%U A163845 3174156512865,16392351740045,84448387609475,434142126555125,2227861180841895,11414655603043335,58403793025471605
%N A163845 Row sums of triangle A163842.
%H A163845 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163845 a(n) = Sum_{k=0..n} Sum_{i=k..n} binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).
%p A163845 swing := proc(n) option remember; if n = 0 then 1 elif
%p A163845 irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p A163845 a := proc(n) local i,k; add(add(binomial(n-k,n-i)*swing(2*i+1),i=k..n),k=0..n) end:
%t A163845 swing[n_] := n! / Floor[n/2]!^2; a[n_] := Sum[Binomial[n-k, n-i] * swing[2*i+1], {k, 0, n}, {i, k, n}]; Array[a, 30, 0] (* _Amiram Eldar_, Aug 22 2025 *)
%Y A163845 Cf. A056040, A163842.
%K A163845 nonn
%O A163845 0,2
%A A163845 _Peter Luschny_, Aug 06 2009