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A334623 Sum of the n-th powers of the descent set statistics for permutations of [n].

%I A334623 #25 Dec 20 2020 12:36:37
%S A334623 1,1,2,18,1576,2675100,128235838496,265039489112493900,
%T A334623 31306198216486969509375104,278983981168082455883720325976751040,
%U A334623 235157286166918393786165504356030195355598048512,23075317400822150539572583950910707053701314350537805923757600
%N A334623 Sum of the n-th powers of the descent set statistics for permutations of [n].
%H A334623 Vaclav Kotesovec, <a href="/A334623/b334623.txt">Table of n, a(n) for n = 0..24</a> (terms 0..21 from Alois P. Heinz)
%H A334623 R. Ehrenborg and A. Happ, <a href="https://arxiv.org/abs/1709.00778">On the powers of the descent set statistic</a>, arXiv:1709.00778 [math.CO], 2017.
%F A334623 a(n) = A334622(n,n).
%F A334623 a(n) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^n.
%p A334623 b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
%p A334623       add(b(u-j, o+j-1, t+1)*x^floor(2^(t-1)), j=1..u)+
%p A334623       add(b(u+j-1, o-j, t+1), j=1..o)))
%p A334623     end:
%p A334623 a:= n-> (p-> add(coeff(p, x, i)^n, i=0..degree(p)))(b(n, 0$2)):
%p A334623 seq(a(n), n=0..12);
%t A334623 b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1,
%t A334623     Sum[b[u - j, o + j - 1, t + 1]*x^Floor[2^(t - 1)], {j, 1, u}] +
%t A334623     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]];
%t A334623 a[n_] := Function[p, Sum[Coefficient[p, x, i]^n, {i, 0, Exponent[p, x]}]][ b[n, 0, 0]];
%t A334623 a /@ Range[0, 12] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)
%Y A334623 Main diagonal of A334622.
%Y A334623 Cf. A060351, A335546.
%K A334623 nonn
%O A334623 0,3
%A A334623 _Alois P. Heinz_, Sep 09 2020