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A362925 Triangle read by rows: T(n,m), n >= 0, 0 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

%I A362925 #36 Jul 09 2025 05:02:10
%S A362925 1,1,1,2,2,1,5,5,4,1,15,15,13,8,1,52,52,47,35,16,1,203,203,188,153,97,
%T A362925 32,1,877,877,825,706,515,275,64,1,4140,4140,3937,3479,2744,1785,793,
%U A362925 128,1,21147,21147,20270,18313,15177,11002,6347,2315,256,1,115975,115975,111835,102678,88033,68303,45368,23073,6817,512,1
%N A362925 Triangle read by rows: T(n,m), n >= 0, 0 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.
%C A362925 A variant of A113547 and A362924. See those entries for further information.
%H A362925 Alois P. Heinz, <a href="/A362925/b362925.txt">Rows n = 0..140, flattened</a>
%F A362925 Sum_{k=0..n} (k+1) * T(n,k) = A040027(n+1). - _Alois P. Heinz_, Dec 02 2023
%e A362925 Triangle begins:
%e A362925        1;
%e A362925        1,      1;
%e A362925        2,      2,      1;
%e A362925        5,      5,      4,      1;
%e A362925       15,     15,     13,      8,     1;
%e A362925       52,     52,     47,     35,    16,     1;
%e A362925      203,    203,    188,    153,    97,    32,     1;
%e A362925      877,    877,    825,    706,   515,   275,    64,     1;
%e A362925     4140,   4140,   3937,   3479,  2744,  1785,   793,   128,    1;
%e A362925    21147,  21147,  20270,  18313, 15177, 11002,  6347,  2315,  256,   1;
%e A362925   115975, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1;
%e A362925   ...
%p A362925 T:= (n, k)-> add(Stirling2(n-k, j)*(j+1)^k, j=0..n-k):
%p A362925 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Dec 01 2023
%t A362925 A362925[n_, m_]:=Sum[StirlingS2[n-m,k](k+1)^m,{k,0,n-m}];
%t A362925 Table[A362925[n,m],{n,0,15},{m,0,n}] (* _Paolo Xausa_, Dec 04 2023 *)
%Y A362925 Row sums are A000110(n+1).
%Y A362925 Columns k=0+1,2-5 give A000110, A078468(n-2) (for n>=2), A383052(n-3) (for n>=3), A383053(n-4) (for n>=4), A383054(n-5) (for n>=5).
%Y A362925 T(n+j,n) give (for j=0-2): A000012, A000079, A007689.
%Y A362925 T(2n,n) gives A367820.
%Y A362925 Cf. A040027, A113547, A362924.
%K A362925 nonn,tabl
%O A362925 0,4
%A A362925 _N. J. A. Sloane_, Aug 10 2023, based on an email from _Don Knuth_