A101842 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

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

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.

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

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

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

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 *)

PROG

(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

CROSSREFS

Cf. A101845, A102012, A011818, A173018.