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 A307663 #46 Jul 30 2024 04:23:30
%S A307663 1,6,41,329,3090,33654,420792,5981688,95782320,1712555280,33909364800,
%T A307663 737868052800,17521164259200,451126883894400,12522623670144000,
%U A307663 372847351488998400,11853064556660275200,400718191717647820800,14354714544806716416000,543129329390299739136000,21642934280974058207232000
%N A307663 a(n) = (n-1)!*(Sum_{i=1..n} Sum_{j=1..i} binomial(i,j)*i/j).
%H A307663 Sela Fried, <a href="/A307663/a307663.pdf">A307663</a>, 2024.
%F A307663 Conjectures from _Robert Israel_, Oct 26 2020: (Start)
%F A307663 E.g.f. ((4*x^2 - 8*x + 5)*log(-x + 1))/(2*(x - 1)^2) - ((4*x^2 - 8*x + 5)*log(1 - 2*x))/(2*(x - 1)^2) + x*(-6 + 5*x)/(4*(x - 1)^2).
%F A307663 D-finite with recurrence 2*(n+3)*(n+2)*n*(n-2)*a(n) - (n+3)*(5*n^2-6*n-17)*a(n+1) + (4 n^2-n-29)* a(n+2) -(n-3)*a(n+3) = 0. (End)
%F A307663 The conjecture regarding the e.g.f. is true. See links. - _Sela Fried_, Jul 30 2024.
%e A307663 a(2) = 1! * (C(1,1)*1/1 + C(2,1)*2/1 + C(2,2)*2/2) = 6.
%t A307663 Array[(# - 1)!*Sum[Sum[Binomial[i, j] i/j, {j, i}], {i, #}] &, 21] (* _Michael De Vlieger_, Apr 21 2019 *)
%o A307663 (PARI) a(n) = (n-1)!*sum(i=1, n, sum(j=1, i, binomial(i,j)*i/j)); \\ _Michel Marcus_, Apr 20 2019
%K A307663 nonn
%O A307663 1,2
%A A307663 _Pedro Caceres_, Apr 20 2019