A369322 - OEIS

%I #17 Jan 26 2024 07:57:23

%S 1,1,3,20,285,8498,521549,65149296,16446593964,8354292354562,

%T 8517018874559019,17400156347544892896,71175200852044807325678,

%U 582639858848549658827324726,9542182685892187892079287210803,312611431819035281373960038697247872

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

%C Column sums of A369321.

%C 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 Alois P. Heinz, <a href="/A369322/b369322.txt">Table of n, a(n) for n = 0..50</a>

%H 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 b:= proc(n, i, t, k) option remember;

%p      `if`(k<0, 0,  `if`(n=0, `if`(k=0, 1, 0), add((d->

%p         b(n-1, j, t+d, k-d))(`if`(j>=i, 1, 0)), j=0..t+1)))

%p     end:

%p a:= n-> add(b(j, -1$2, n), j=n..n*(n+1)/2):

%p seq(a(n), n=0..15);  # _Alois P. Heinz_, Jan 25 2024

%Y Cf. A369321, A336070.

%K nonn

%O 0,3

%A _Joerg Arndt_, Jan 20 2024