

A101842


Triangle read by rows: T(n,k), for k=n..n1, is the scaled (by 2^n n!) probability that the sum of n uniform [1, 1] variables is between k and k+1.


3



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

1,4


COMMENTS

Equivalently, T(n,k)/n! is the ndimensional volume of the portion of the ndimensional hypercube [1,1]^n cut by the (n1)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 flagdescent (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 JosuatVergès, Apr 25 2011]
T(n,n) = A011818(n), the normalized volume of center slice of ndimensional 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), 210224.
R. Stanley, Eulerian partitions of a unit hypercube, in Higher Combinatorics (M. Aigner, ed.), Reidel, Dordrecht/Boston, 1977, p. 49.


FORMULA

T(n,k) = (nk) * T(n1,k1) + T(n1,k) + (n+k+1)*T(n1,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,ki), n >= 1.  Peter Bala, Jul 17 2012
E.g.f.: 1 + (1y) / (1  y*exp( (1y^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..n1 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 <= n1 := T[n, k] = (nk) T[n1, k1] + T[n1, k] + (n+k+1) T[n1, k+1]; T[1, 1] = T[1, 0] = 1; T[_, _] = 0;
Table[T[n, k], {n, 1, 8}, {k, n, n1}] (* JeanFrançois Alcover, Jun 27 2019 *)


PROG

(PARI) {T(n, k) = polcoeff( n!*polcoeff( 1 + (1y) / (1  y*exp( (1y^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.
See also A008292 (Eulerian numbers).
Sequence in context: A196601 A196578 A196809 * A196786 A196746 A196904
Adjacent sequences: A101839 A101840 A101841 * A101843 A101844 A101845


KEYWORD

nonn,tabf


AUTHOR

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


EXTENSIONS

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


STATUS

approved



