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This sequence starts for k and n with offset 1. If it would start with k = 0, we would observe in column k = 0 an exact copy of column k = 1 with a preceding one at n = 0, k = 0. The difference between 0 ranks and one rank (all in the same rank) is only for n = 0 where k = 0 allows zero-filled ranks as an valid arrangement, too. (End)
(End)
proposed
editing
editing
proposed
This sequence starts for k and n with offset 1. If it would start with k = 0, we would observe in column k = 0 an exact copy of column k = 1 with a preceeding preceding one at n = 0, k = 0. The difference between 0 ranks and one rank (all in the same rank) is only for n = 0 where k = 0 allows zero-filled ranks as an valid arrangement, too.
E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). -_ _Vladeta Jovovic_, Sep 14 2003
E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). - _Vladeta Jovovic, _, Sep 14 2003
T(n, k) = Sum_{m = k..n} A090582(n + 1, m + 1).
From Thomas Scheuerle, Apr 25 2022: (Start)
Sum_{k = 0..n} T(n, k) = A005649(n). Column k = 0 i not part of data.
Sum_{k = 1..n} T(n, k) = A005649(n).
T(n, 0) = A000670(n). Column k = 0 i not part of data.
T(n, 1) = A000670(n), for n > 0.
T(n, 2) = A052875(n).
T(n, 3) = A102232(n).
T(n, n) = n! = A000142. (End)
E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). - Vladeta Jovovic, Sep 14 2003