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A369322 a(n) is the number of weak ascent sequences (of any length) with n weak ascents.

%I A369322 #21 Jul 07 2025 10:21:22
%S A369322 1,1,3,20,285,8498,521549,65149296,16446593964,8354292354562,
%T A369322 8517018874559019,17400156347544892896,71175200852044807325678,
%U A369322 582639858848549658827324726,9542182685892187892079287210803,312611431819035281373960038697247872
%N A369322 a(n) is the number of weak ascent sequences (of any length) with n weak ascents.
%C A369322 Column sums of A369321.
%C A369322 A weak ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) counts the weak ascents d(j) >= d(j-1) of its argument.
%H A369322 Alois P. Heinz, <a href="/A369322/b369322.txt">Table of n, a(n) for n = 0..50</a>
%H A369322 Beata Benyi, Anders Claesson, Mark Dukes, <a href="https://arxiv.org/abs/2111.03159">Weak ascent sequences and related combinatorial structures</a>, arXiv:2111.03159 [math.CO], (4-November-2021).
%p A369322 b:= proc(n, i, t, k) option remember;
%p A369322      `if`(k<0, 0,  `if`(n=0, `if`(k=0, 1, 0), add((d->
%p A369322         b(n-1, j, t+d, k-d))(`if`(j>=i, 1, 0)), j=0..t+1)))
%p A369322     end:
%p A369322 a:= n-> add(b(j, -1$2, n), j=n..n*(n+1)/2):
%p A369322 seq(a(n), n=0..15);  # _Alois P. Heinz_, Jan 25 2024
%t A369322 b[n_, i_, t_, k_] := b[n, i, t, k] = If[k < 0, 0, If[n == 0, If[k == 0, 1, 0], Sum[Function[d, b[n-1, j, t+d, k-d]][If[j >= i, 1, 0]], {j, 0, t+1}]]];
%t A369322 a[n_] := Sum[b[j, -1, -1, n], {j, n, n*(n+1)/2}];
%t A369322 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jul 07 2025, after _Alois P. Heinz_ *)
%Y A369322 Cf. A369321, A336070.
%K A369322 nonn
%O A369322 0,3
%A A369322 _Joerg Arndt_, Jan 20 2024