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A347930 3-Springer numbers.

%I A347930 #13 Sep 20 2021 09:40:07
%S A347930 1,1,3,16,88,625,5527,55760,640540,8329326,120212331,1905939913,
%T A347930 32987637967,618591571085,12489644875037,270193806214360,
%U A347930 6235154917414954,152875655211527878,3968729594485785289,108754865309750398187,3137052120203959610759
%N A347930 3-Springer numbers.
%C A347930 a(n) is also the volume of a certain flow polytope.
%H A347930 Arvind Ayyer, Matthieu Josuat-Vergès, and Sanjay Ramassamy, <a href="https://doi.org/10.5802/ahl.33">Extensions of partial cyclic orders and consecutive coordinate polytopes</a>, Ann. H. Lebesgue, 3 (2020), 275-297.
%H A347930 R. S. Gonzalez D'Leon, A. H. Morales, C. R. H. Hanusa, and M. Yip, <a href="https://arxiv.org/abs/2107.07326">Column convex matrices, G-cyclic orders, and flow polytopes</a>, arXiv:2107.07326 [math.CO], 2021.
%H A347930 S. Ramassamy, <a href="https://doi.org/10.37236/7145">Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons</a>, Electron. J. Combin., 25 (2018), #P1.66.
%F A347930 a(n) = Sum_{(x,y,z), x+y+z=n-2} ((n-2)!/(x!*y!*z!))*b(x,y,z), where b(x,y,z) are the 3-Entringer numbers defined by Ramassamy.
%p A347930 wcomps:=proc(n,k)
%p A347930        option remember;
%p A347930 local ocomps,ncomps,i;
%p A347930 ocomps:=combinat:-composition(n+k,k);
%p A347930 ncomps:={};
%p A347930 for i from 1 to nops(ocomps) do
%p A347930    ncomps:=ncomps union{[seq(ocomps[i][j]-1,j=1..k)]};
%p A347930 end do;
%p A347930 return [op(ncomps)];
%p A347930 end proc:
%p A347930 b:=proc(s) option remember;
%p A347930    local k;
%p A347930    k := nops(s);
%p A347930    if s = [seq(0,i=1..k)] then
%p A347930       return(1);
%p A347930    elif s[1]>0 then
%p A347930       return(add(b([s[2]+j,op(s[3..k]),s[1]-j-1]),j=0..s[1]-1));
%p A347930    else
%p A347930       return(0);
%p A347930    end if;
%p A347930 end proc:a:=proc(n)   local N,S:   N := n-2;   S := wcomps(N,3);   return add(combinat:-multinomial(N,op(s))*b(s), s in S);end proc:seq(a(n),n=2..10);
%Y A347930 Cf. A001586, A008282, A000111.
%K A347930 nonn
%O A347930 2,3
%A A347930 _Alejandro H. Morales_, Sep 19 2021