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

1, 1, 3, 20, 285, 8498, 521549, 65149296, 16446593964, 8354292354562, 8517018874559019, 17400156347544892896, 71175200852044807325678, 582639858848549658827324726, 9542182685892187892079287210803, 312611431819035281373960038697247872

Offset: 0

Author

Joerg Arndt, Jan 20 2024

Keywords

• nonn

Comments

Column sums of A369321.

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50
• Beata Benyi, Anders Claesson, Mark Dukes, Weak ascent sequences and related combinatorial structures, arXiv:2111.03159 [math.CO], (4-November-2021).

Crossrefs

Cf. A369321, A336070.

Programs

Maple

b:= proc(n, i, t, k) option remember;
     `if`(k<0, 0,  `if`(n=0, `if`(k=0, 1, 0), add((d->
        b(n-1, j, t+d, k-d))(`if`(j>=i, 1, 0)), j=0..t+1)))
    end:
a:= n-> add(b(j, -1$2, n), j=n..n*(n+1)/2):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 25 2024

Mathematica

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}]]];
a[n_] := Sum[b[j, -1, -1, n], {j, n, n*(n+1)/2}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 07 2025, after Alois P. Heinz *)