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A059298
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Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 2.
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2
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1, 2, 1, 3, 6, 1, 4, 24, 12, 1, 5, 80, 90, 20, 1, 6, 240, 540, 240, 30, 1, 7, 672, 2835, 2240, 525, 42, 1, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 10, 11520, 262440, 860160, 787500, 272160, 41160
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The inverse triangle is the signed version 1,-2,1,9,-6,1,.. of triangle A061356. [From Peter Luschny (peter(AT)luschny.de), Mar 13 2009]
T(n,k) is the sum of the products of the cardinality of the blocks (cells) in the set partitions of {1,2,..,n} into exactly k blocks.
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
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EXAMPLE
| Triangle begins 1; 0, 1; 0, 2, 1; 0, 3, 6, 1; 0, 4, 24, 12, 1; ...
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MATHEMATICA
| t = Transpose[ Table[ Range[0, 11]! CoefficientList[ Series[(x Exp[x])^n/n!, {x, 0, 11}], x], {n, 11}]]; Table[ t[[n, k]], {n, 2, 11}, {k, n - 1}] // Flatten (* or simply *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Table[t[n, k], {n, 10}, {k, n}] // Flatten
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CROSSREFS
| There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.
Sequence in context: A115597 A103371 A120257 * A156914 A059434 A182928
Adjacent sequences: A059295 A059296 A059297 * A059299 A059300 A059301
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 25 2001
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