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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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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, 1, 255, 6297, 56935, 274677
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OFFSET
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1,4
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COMMENTS
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Equivalently, T(n,k)/n! is the n-dimensional volume of the portion of the n-dimensional hypercube [-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]
T(n,n) = A011818(n), the normalized volume of center slice of n-dimensional cube. - Paul D. Hanna, Jan 03 2017
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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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LINKS
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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).
With the column index k starting at 0 the table entries appear to be given by t(n,k) = Sum_{i = 0..k} A173018(n,i)*binomial(n,k-i), n >= 1. - Peter Bala, Jul 17 2012
E.g.f.: -1 + (1-y) / (1 - y*exp( (1-y^2)*x )). - Paul D. Hanna, Jan 03 2017
p(n, x) = n! * (x - 1)^n * [z^n] (1 - x)/(exp((x + 1)*z) - x) defines a sequence of polynomials. p(n, x) = (x + 1)^n*A(n, x) where A(n, x) are the Eulerian polynomials as defined by A008292. The coefficients of these polynomials are the T(n, k) if enumerated for n >= 0 and 0 <= k <= 2*n - 1 if n > 0 and T(0,0) = 1. - Peter Luschny, Jun 24 2019
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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
The sequence of polynomials starts:
p(0, x) = 1
p(1, x) = x + 1
p(2, x) = x^3 + 3*x^2 + 3*x + 1
p(3, x) = x^5 + 7*x^4 + 16*x^3 + 16*x^2 + 7*x + 1
p(4, x) = x^7 + 15*x^6 + 61*x^5 + 115*x^4 + 115*x^3 + 61*x^2 + 15*x + 1
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MAPLE
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gf := (1 - x)/(exp((x + 1)*z) - x): ser := series(gf, z, 12):
for n from 0 to 6 do p := simplify(n!*(x - 1)^n*coeff(ser, z, n));
print(PolynomialTools:-CoefficientList(p, x)) od: # Peter Luschny, Jun 24 2019
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MATHEMATICA
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T[n_, k_] /; -n <= k <= n-1 := T[n, k] = (n-k) T[n-1, k-1] + T[n-1, k] + (n+k+1) T[n-1, k+1]; T[1, -1] = T[1, 0] = 1; T[_, _] = 0;
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PROG
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(PARI) {T(n, k) = polcoeff( n!*polcoeff( -1 + (1-y) / (1 - y*exp( (1-y^2)*x +x*O(x^n) )) , n, x), k, y)}
for(n=1, 10, for(k=1, 2*n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 03 2017
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CROSSREFS
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See also A008292 (Eulerian numbers).
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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