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%I #11 Sep 01 2021 17:10:39
%S 1,1,3,19,258,7406,442668,54371100,13585980916,6859762797636,
%T 6969135518632452,14209819222900305044,58061006907633910998660,
%U 474996314819118381967232244,7776635831062534849079443379908,254723669580125156112963535996038036
%N Total number of permutations p of [k] such that n is the maximum of the partial sums of the signed up-down jump sequence of 0,p summed over all k >= 0.
%C An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump -j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.
%H Alois P. Heinz, <a href="/A316294/b316294.txt">Table of n, a(n) for n = 0..40</a>
%p b:= proc(u, o, c, k) option remember;
%p `if`(c>k, 0, `if`(u+o=0, 1,
%p add(b(u-j, o-1+j, c+j, k), j=1..u)+
%p add(b(u+j-1, o-j, c-j, k), j=1..o)))
%p end:
%p a:= n-> add(b(k, 0$2, n)-b(k, 0$2, n-1), k=n..n*(n+1)/2):
%p seq(a(n), n=0..15);
%t b[u_, o_, c_, k_] := b[u, o, c, k] =
%t If[c > k, 0, If[u + o == 0, 1,
%t Sum[b[u - j, o - 1 + j, c + j, k], {j, u}] +
%t Sum[b[u + j - 1, o - j, c - j, k], {j, o}]]];
%t a[n_] := Sum[b[k, 0, 0, n] - b[k, 0, 0, n-1], {k, n, n(n+1)/2}];
%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Sep 01 2021, after _Alois P. Heinz_ *)
%Y Column sums of A316292 or A316293.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jun 28 2018
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