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Search: id:A101842
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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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+0
3
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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; listen)
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OFFSET
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1,4
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COMMENT
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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.
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REFERENCES
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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
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FORMULA
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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
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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
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Cf. A101845, A102012. See also A008292 (Eulerian numbers).
Sequence in context: A075772 A142157 A119608 this_sequence A165795 A099037 A104378
Adjacent sequences: A101839 A101840 A101841 this_sequence A101843 A101844 A101845
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KEYWORD
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nonn,tabf
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AUTHOR
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David Applegate (david(AT)research.att.com), based on Peter Doyle's talk, Jun 10 2007
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