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A179457
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Triangle read by rows: number of permutation trees of power n and width <= k.
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2
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1, 1, 2, 1, 5, 6, 1, 12, 23, 24, 1, 27, 93, 119, 120, 1, 58, 360, 662, 719, 720, 1, 121, 1312, 3728, 4919, 5039, 5040, 1, 248, 4541, 20160, 35779, 40072, 40319, 40320, 1, 503, 15111, 103345, 259535, 347769, 362377, 362879, 362880
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OFFSET
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1,3
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COMMENTS
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Partial row sums of A008292 (triangle of Eulerian numbers).
Given by a very similar formula.
Special case: A179457(n,2) = A000325(n) for n > 1 (Grassmannian permutations).
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 533.
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LINKS
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FORMULA
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T(n,k) = sum( ((-1)^j*(k-j)^(n+1))*binomial(n+1,j),j=0..k) - Olivier Gérard, Aug 04 2012
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EXAMPLE
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1;
1, 2;
1, 5, 6;
1, 12, 23, 24;
1, 27, 93, 119, 120;
1, 58, 360, 662, 719, 720;
1, 121, 1312, 3728, 4919, 5039, 5040;
1, 248, 4541, 20160, 35779, 40072, 40319, 40320;
1, 503, 15111, 103345, 259535, 347769, 362377, 362879, 362880;
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MAPLE
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Eulerian:= (n, k)-> sum((-1)^j*(k-j+1)^n * binomial(n+1, j), j=0..k+1):
s:=(j, n)-> sum(Eulerian(j, k-1), k=1..n):
for i from 1 to 15 do print(seq(s(i, n), n=1..i)) od; # Gary Detlefs, Nov 18 2011
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MATHEMATICA
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Flatten[Table[Table[Sum[(-1)^j (k-j)^(n+1) Binomial[n+1, j], {j, 0, k}], {k, 1, n + 1}], {n, 0, 10}], 1] (* Olivier Gérard, Aug 04 2012 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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