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A154715
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Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).
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0
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1, 2, 3, 4, 18, 16, 8, 81, 192, 125, 16, 324, 1536, 2500, 1296, 32, 1215, 10240, 31250, 38880, 16807, 64, 4374, 61440, 312500, 699840, 705894, 262144, 128, 15309, 344064, 2734375, 9797760, 17294403, 14680064, 4782969
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Formatted as a square array:
1th row is A000079(n). Subsets of an n-set.
2nd row is A036290(n+1). Special (n+1)-subsets of a 3n-set partitioned into 3-blocks.
2nd column is A066274(n+1). Endofunctions of [n] such that 1 is not a fixed point.
1th column is A000272(n+2). Trees on n labeled nodes (Cayley's formula).
Alternating sum of rows in the triangle, Sum(k=0..n, (-1)^(n-k)T(n,k)) = n! (A000142(n)).
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LINKS
| Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets, web
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FORMULA
| T(n,k) = C(n,k)*(k+2)^n where C(n,k) is the binomial coefficient (A007318) and n>=0 and k>=0.
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EXAMPLE
| 1
2, 3
4, 18, 16
8, 81, 192, 125
16, 324, 1536, 2500, 1296
32, 1215, 10240, 31250, 38880, 16807
64, 4374, 61440, 312500, 699840, 705894, 262144
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MAPLE
| T := proc(n, k) binomial(n, k)*(k+2)^n end;
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CROSSREFS
| Ref. A000079, A000272, A036290, A066274.
Sequence in context: A037394 A037430 A187075 * A077407 A123702 A067805
Adjacent sequences: A154712 A154713 A154714 * A154716 A154717 A154718
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Jan 14 2009
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