cp's OEIS Frontend

A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.

%I A084416 #25 Jun 17 2022 15:58:20
%S A084416 1,3,2,13,12,6,75,74,60,24,541,540,510,360,120,4683,4682,4620,4080,
%T A084416 2520,720,47293,47292,47166,45360,36960,20160,5040,545835,545834,
%U A084416 545580,539784,498960,372960,181440,40320,7087261,7087260,7086750,7068600,6882120,6048000,4142880,1814400,362880
%N A084416 Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.
%C A084416 Interpolates between A000670 and factorials.
%C A084416 From _Thomas Scheuerle_, Apr 25 2022: (Start)
%C A084416 Number of preferential arrangements of n labeled elements when at least k ranks are required.
%C A084416 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)
%F A084416 E.g.f. for m-th column: (exp(x)-1)^m/(2-exp(x)). - _Vladeta Jovovic_, Sep 14 2003
%F A084416 T(n, k) = Sum_{m = k..n} A090582(n + 1, m + 1).
%F A084416 From _Thomas Scheuerle_, Apr 25 2022: (Start)
%F A084416 Sum_{k = 0..n} T(n, k) = A005649(n). Column k = 0 is not part of data.
%F A084416 Sum_{k = 1..n} T(n, k) = A069321(n).
%F A084416 T(n, 0) = A000670(n). Column k = 0 is not part of data.
%F A084416 T(n, 1) = A000670(n), for n > 0.
%F A084416 T(n, 2) = A052875(n).
%F A084416 T(n, 3) = A102232(n).
%F A084416 T(n, n) = n! = A000142. (End)
%e A084416 Triangle begins with T(n,k):
%e A084416    k=   1,   2,   3,   4,   5
%e A084416   n=1   1
%e A084416   n=2   3,   2
%e A084416   n=3  13,  12,   6
%e A084416   n=4  75,  74,  60,  24
%e A084416   n=5 541, 540, 510, 360, 120
%e A084416 ...
%e A084416 From _Thomas Scheuerle_, Apr 25 2022: (Start)
%e A084416 If we would add n = 0, k = 0 to the data of this sequence:
%e A084416    k=   0,   1,   2,
%e A084416   n=0   1
%e A084416   n=1   1,   1
%e A084416   n=2   3,   3,   2
%e A084416 ...
%e A084416 T(n, 3) with 3 preceding zeros is: 0,0,0,6,60,510,4620,...
%e A084416 This sequence has the e.g.f.: (e^x-1)^3/(2-e^x).
%e A084416 .
%e A084416 13 arrangements for n = 3 and k = 1 (one rank required):
%e A084416 1,2,3  1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
%e A084416 12 arrangements for n = 3 and k = 2 (two ranks required):
%e A084416 1,2|3  2,3|1  1,3|2  1|2,3  2|1,3  3|1,2  1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
%e A084416 6 arrangements for n = 3 and k = 3 (three ranks required):
%e A084416 1|2|3  1|3|2  2|1|3  2|3|1  3|1|2  3|2|1
%e A084416 . (End)
%p A084416 T := (n,k)->sum(i!*Stirling2(n,i),i=k..n): seq(seq(T(n,k),k=1..n),n=1..10);
%o A084416 (PARI) row(n) = vector(n, k, sum(i=k, n, i!*stirling(n, i, 2))); \\ _Michel Marcus_, Apr 20 2022
%Y A084416 Mirror image of array in A084417.
%Y A084416 Cf. A005649, A069321 (row sums).
%Y A084416 A000670(n) (column k = 1), A052875(n) (column k = 2), A102232(n) (column k = 3).
%Y A084416 Cf. A000142, A008277.
%K A084416 nonn,tabl,easy
%O A084416 1,2
%A A084416 _N. J. A. Sloane_, Jun 24 2003
%E A084416 More terms from _Emeric Deutsch_, May 11 2004
%E A084416 More terms from _Michel Marcus_, Apr 20 2022