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

%I A320964 #31 May 17 2023 08:54:35
%S A320964 1,1,2,3,5,9,18,40,98,262,757,2344,7723,26918,98790,380361,1531699,
%T A320964 6434386,28130891,127729731,601196429,2928369918,14738842362,
%U A320964 76547694742,409718539682,2257459567237,12789959138944,74439150889081,444647798089246,2723583835351856
%N A320964 a(n) = Sum_{j=0..n} Sum_{k=0..j} Stirling2(j - k, k).
%C A320964 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.
%C A320964 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
%H A320964 Alois P. Heinz, <a href="/A320964/b320964.txt">Table of n, a(n) for n = 0..665</a>
%p A320964 ListTools:-PartialSums([seq(add(Stirling2(n-k, k), k=0..n), n=0..29)]);
%p A320964 # second Maple program:
%p A320964 b:= proc(n, m) option remember; `if`(n>m,
%p A320964       b(n-1, m)*m+b(n-1, m+1), `if`(n=m, 1, 0))
%p A320964     end:
%p A320964 a:= proc(n) a(n):= `if`(n=0, 0, a(n-1))+b(n, 0) end:
%p A320964 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 16 2023
%t A320964 a[n_] := Sum[Sum[StirlingS2[j - k, k], {k, 0, j}], {j, 0, n}]; Array[a, 30, 0] (* _Amiram Eldar_, Nov 06 2018 *)
%t A320964 Table[Sum[StirlingS2[j-k,k],{j,0,n},{k,0,j}],{n,0,30}] (* _Harvey P. Dale_, May 15 2019 *)
%o A320964 (PARI) a(n)={sum(j=0, n, sum(k=0, j, abs(stirling(j-k, k, 2))))} \\ _Andrew Howroyd_, Nov 06 2018
%Y A320964 Row sums of A320955 seen as a triangle.
%Y A320964 Cf. A171367, A048993.
%K A320964 nonn
%O A320964 0,3
%A A320964 _Peter Luschny_, Nov 06 2018