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A173018
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Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
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20
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1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, 1, 0, 1, 26, 66, 26, 1, 0, 1, 57, 302, 302, 57, 1, 0, 1, 120, 1191, 2416, 1191, 120, 1, 0, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0, 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1, 0
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graph;
refs;
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history;
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OFFSET
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0,8
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COMMENTS
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This version indexes the Eulerian numbers in the same way as Graham et al.'s Concrete Mathematics. The traditional indexing, used by Riordan, Comtet and others, is given in A008292, which is the main entry for the Eulerian numbers.
Each row of A123125 is the reverse of the corresponding row in A173018. - Michael Somos Mar 17 2011
Triangle T(n,k), read by rows, given by [1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...] DELTA [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...] where DELTA is the operator defined in A084938. - Philippe DELEHAM Sep 30 2011
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REFERENCES
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Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, Arxiv preprint arXiv:1105.3043, 2011
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 254.
P. Hitczenko and S. Janson, Weighted random staircase tableaux, arXiv preprint arXiv:1212.5498, 2012. - From N. J. A. Sloane, Feb 02 2013
See also A008292.
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LINKS
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Alois P. Heinz, Rows n = 0..140, flattened
Digital Library of Mathematical Functions, Permutations: Order Notation
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EXAMPLE
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Triangle begins:
[ 0] 1,
[ 1] 1, 0,
[ 2] 1, 1, 0,
[ 3] 1, 4, 1, 0,
[ 4] 1, 11, 11, 1, 0,
[ 5] 1, 26, 66, 26, 1, 0,
[ 6] 1, 57, 302, 302, 57, 1, 0,
[ 7] 1, 120, 1191, 2416, 1191, 120, 1, 0,
[ 8] 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0,
[ 9] 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0,
[10] 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1, 0,
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MAPLE
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T:= proc(n, k) option remember;
if k=0 and n>=0 then 1
elif k<0 or k>n then 0
else (n-k) * T(n-1, k-1) + (k+1) * T(n-1, k)
fi
end:
seq (seq (T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Jan 14 2011
# Maple since version 13:
A173018 := (n, k) -> combinat[eulerian1](n, k): # Peter Luschny, Nov 11 2012
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MATHEMATICA
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t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0;
t[n_, k_] := t[n, k] = (n-k)*t[n-1, k-1] + (k+1)*t[n-1, k]; Flatten[Table[t[n, k], {n, 0, 11}, {k, 0, n}]][[1 ;; 60]]
(* From Jean-François Alcover, Apr 29 2011, after Maple program *)
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PROG
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(Sage)
@CachedFunction
def eulerian1(n, k):
if k==0: return 1
if k==n: return 0
return eulerian1(n-1, k)*(k+1)+eulerian1(n-1, k-1)*(n-k)
for n in (0..9): [eulerian1(n, k) for k in(0..n)] # Peter Luschny, Nov 11 2012
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CROSSREFS
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Cf. A008292. A123125 is a less satisfactory variant. A008517 and A201637 for the second-order numbers.
Sequence in context: A086329 A085852 A123125 * A055105 A200545 A058710
Adjacent sequences: A173015 A173016 A173017 * A173019 A173020 A173021
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, Nov 21 2010
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STATUS
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approved
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