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A084416
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Triangle read by rows: T(n,k) = Sum_{i=k..n} i!*Stirling2(n,i), n >= 1, 1 <= k <= n.
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4
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1, 3, 2, 13, 12, 6, 75, 74, 60, 24, 541, 540, 510, 360, 120, 4683, 4682, 4620, 4080, 2520, 720, 47293, 47292, 47166, 45360, 36960, 20160, 5040, 545835, 545834, 545580, 539784, 498960, 372960, 181440, 40320, 7087261, 7087260, 7086750, 7068600, 6882120, 6048000, 4142880, 1814400, 362880
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OFFSET
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1,2
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COMMENTS
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Interpolates between A000670 and factorials.
Number of preferential arrangements of n labeled elements when at least k ranks are required.
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)
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LINKS
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FORMULA
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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).
Sum_{k = 0..n} T(n, k) = A005649(n). Column k = 0 is not part of data.
Sum_{k = 1..n} T(n, k) = A069321(n).
T(n, 0) = A000670(n). Column k = 0 is not part of data.
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EXAMPLE
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Triangle begins with T(n,k):
k= 1, 2, 3, 4, 5
n=1 1
n=2 3, 2
n=3 13, 12, 6
n=4 75, 74, 60, 24
n=5 541, 540, 510, 360, 120
...
If we would add n = 0, k = 0 to the data of this sequence:
k= 0, 1, 2,
n=0 1
n=1 1, 1
n=2 3, 3, 2
...
T(n, 3) with 3 preceding zeros is: 0,0,0,6,60,510,4620,...
This sequence has the e.g.f.: (e^x-1)^3/(2-e^x).
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13 arrangements for n = 3 and k = 1 (one rank required):
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
12 arrangements for n = 3 and k = 2 (two ranks required):
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
6 arrangements for n = 3 and k = 3 (three ranks required):
1|2|3 1|3|2 2|1|3 2|3|1 3|1|2 3|2|1
. (End)
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MAPLE
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T := (n, k)->sum(i!*Stirling2(n, i), i=k..n): seq(seq(T(n, k), k=1..n), n=1..10);
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PROG
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(PARI) row(n) = vector(n, k, sum(i=k, n, i!*stirling(n, i, 2))); \\ Michel Marcus, Apr 20 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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