A320964 a(n) = Sum_{j=0..n} Sum_{k=0..j} Stirling2(j - k, k).

1, 1, 2, 3, 5, 9, 18, 40, 98, 262, 757, 2344, 7723, 26918, 98790, 380361, 1531699, 6434386, 28130891, 127729731, 601196429, 2928369918, 14738842362, 76547694742, 409718539682, 2257459567237, 12789959138944, 74439150889081, 444647798089246, 2723583835351856

Offset: 0

Author

Peter Luschny, Nov 06 2018

Keywords

• nonn

Comments

The row sums of A320955 seen as a triangle are the partial sums of the antidiagonal sums of the triangle of the Stirling set numbers.

Number of partitions of [n] into m blocks that are ordered with increasing least elements and where block m-j contains n-j (m in {0..n}, j in {0..m-1}). a(5) = 9: 12345, 1234|5, 123|4|5, 124|35, 12|3|4|5, 134|25, 13|24|5, 14|235, 1|2|3|4|5. - Alois P. Heinz, May 16 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..665

Crossrefs

Row sums of A320955 seen as a triangle.

Cf. A171367, A048993.

Programs

Maple

ListTools:-PartialSums([seq(add(Stirling2(n-k, k), k=0..n), n=0..29)]);
# second Maple program:
b:= proc(n, m) option remember; `if`(n>m,
      b(n-1, m)*m+b(n-1, m+1), `if`(n=m, 1, 0))
    end:
a:= proc(n) a(n):= `if`(n=0, 0, a(n-1))+b(n, 0) end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2023

Mathematica

a[n_] := Sum[Sum[StirlingS2[j - k, k], {k, 0, j}], {j, 0, n}]; Array[a, 30, 0] (* Amiram Eldar, Nov 06 2018 *)
Table[Sum[StirlingS2[j-k,k],{j,0,n},{k,0,j}],{n,0,30}] (* Harvey P. Dale, May 15 2019 *)

PARI

a(n)={sum(j=0, n, sum(k=0, j, abs(stirling(j-k, k, 2))))} \\ Andrew Howroyd, Nov 06 2018