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a(n) = (n-1)!*(Sum_{i=1..n} Sum_{j=1..i} binomial(i,j)*i/j).
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%I #46 Jul 30 2024 04:23:30

%S 1,6,41,329,3090,33654,420792,5981688,95782320,1712555280,33909364800,

%T 737868052800,17521164259200,451126883894400,12522623670144000,

%U 372847351488998400,11853064556660275200,400718191717647820800,14354714544806716416000,543129329390299739136000,21642934280974058207232000

%N a(n) = (n-1)!*(Sum_{i=1..n} Sum_{j=1..i} binomial(i,j)*i/j).

%H Sela Fried, <a href="/A307663/a307663.pdf">A307663</a>, 2024.

%F Conjectures from _Robert Israel_, Oct 26 2020: (Start)

%F 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 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 The conjecture regarding the e.g.f. is true. See links. - _Sela Fried_, Jul 30 2024.

%e a(2) = 1! * (C(1,1)*1/1 + C(2,1)*2/1 + C(2,2)*2/2) = 6.

%t Array[(# - 1)!*Sum[Sum[Binomial[i, j] i/j, {j, i}], {i, #}] &, 21] (* _Michael De Vlieger_, Apr 21 2019 *)

%o (PARI) a(n) = (n-1)!*sum(i=1, n, sum(j=1, i, binomial(i,j)*i/j)); \\ _Michel Marcus_, Apr 20 2019

%K nonn

%O 1,2

%A _Pedro Caceres_, Apr 20 2019