A101842 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.

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

Offset: 1

Author

David Applegate, based on Peter Doyle's talk, Jun 10 2007

Keywords

• nonn
• tabf

Comments

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

Examples

			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
		

References

• 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

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..2070 for rows 1..45 of flattened triangle
• R. M. Adin, F. Brenti and Y. Roichman, Descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math. 27 (2001), 210-224.
• R. Stanley, Eulerian partitions of a unit hypercube, in Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht/Boston, 1977, p. 49.

Crossrefs

Cf. A101845, A102012, A011818, A173018.

See also A008292 (Eulerian numbers).

Programs

Maple

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

Mathematica

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;
Table[T[n, k], {n, 1, 8}, {k, -n, n-1}] (* Jean-François Alcover, Jun 27 2019 *)

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

Formula

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

Extensions

Terms a(57) and beyond from Paul D. Hanna, Jan 03 2017