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A101842
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Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [ -1,1] variables is between k and k+1.
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2
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1, 1, 1, 3, 3, 1, 1, 7, 16, 16, 7, 1, 1, 15, 61, 115, 115, 61, 15, 1, 1, 31, 206, 626, 1056, 1056, 626, 206, 31, 1, 1, 63, 659, 2989, 7554, 11774, 11774, 7554, 2989, 659, 63, 1, 1, 127, 2052, 13308, 47349, 105099, 154624, 154624, 105099, 47349, 13308, 2052, 127, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Equivalently, T(n,k)/n! is the n-dimensional volume of the portion of the n-dimensional hyper-cube [ -1,1]^n cut by the (n-1)-dimensional hyperplanes x_1 + x_2 + ... x_n = k, x_1 + x_2 + ... x_n = k+1.
The analogous triangle for the interval [0,1] is that of the Eulerian numbers, A008292.
This is the distribution of the flag-descent (fdes) statistic in signed permutations, as introduced by Adin, Brenti, Roichman. The link between fdes and portions of hypercubes follows an argument adapted from Stanley. [Matthieu Josuat-Vergès, Apr 25 2011]
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REFERENCES
| R.M. Adin, F. Brenti and Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math. 27 (2001), 210--224.
Peter Doyle, Myths about card shuffling, talk given at DIMACS Workshop on Puzzling Mathematics and Mathematical Puzzles: a Gathering in Honor of Peter Winkler's 60th Birthday, Rutgers University, Jun 08, 2007
R. Stanley, Eulerian partitions of a unit hypercube, in Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht/Boston, 1977, p. 49.
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FORMULA
| T(n,k) = (n-k) * T(n-1,k-1) + T(n-1,k) + (n+k+1)*T(n-1,k+1).
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EXAMPLE
| Triangle of T(n,k), k=-n..n-1 begins:
.....................................1,.1
..................................1,.3,.3,.1
..............................1,.7,.16,.16,.7,.1
........................1,.15,.61,.115,.115,.61,.15,.1
.................1,.31,.206,.626,.1056,.1056,.626,.206,.31,.1
.........1,.63,.659,.2989,.7554,.11774,.11774,.7554,.2989,.659,.63,.1
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CROSSREFS
| Cf. A101845, A102012. See also A008292 (Eulerian numbers).
Sequence in context: A196601 A196578 A196809 * A196786 A196746 A196904
Adjacent sequences: A101839 A101840 A101841 * A101843 A101844 A101845
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KEYWORD
| nonn,tabf
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AUTHOR
| David Applegate (david(AT)research.att.com), based on Peter Doyle's talk, Jun 10 2007
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