A182822 Exponential Riordan array, defining orthogonal polynomials related to permutations without double falls.

1, 1, 1, 2, 3, 1, 5, 12, 6, 1, 17, 53, 39, 10, 1, 70, 279, 260, 95, 15, 1, 349, 1668, 1914, 880, 195, 21, 1, 2017, 11341, 15330, 8554, 2380, 357, 28, 1, 13358, 86019, 134317, 87626, 29379, 5530, 602, 36, 1, 99377, 722664, 1277604, 954885, 372771, 84231, 11508, 954, 45, 1, 822041, 6655121, 13149441, 11061480, 4924515, 1292445, 211533, 22020, 1440, 55, 1

Offset

0,4

Comments

Inverse is the coefficient array for the orthogonal polynomials P(0,x) = 1, P(1,x) = x-1, P(n,x) = (x-n)*P(n-1,x) - (n-1)^2*P(n-2,x). Inverse is A182823. First column is A049774.

Links

Table of n, a(n) for n=0..65.

Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.

Formula

Exponential Riordan array [exp(x/2)/(cos(sqrt(3)x/2)-sin(sqrt(3)x/2)/sqrt(3)), 2*sin(sqrt(3)x/2)/(sqrt(3)*cos(sqrt(3)x/2)-sin(sqrt(3)x/2))].

From Werner Schulte, Mar 27 2022: (Start)

T(n,k) = T(n-1,k-1) + (k+1) * T(n-1,k) + (k+1)^2 * T(n-1,k+1) for n > 0 with initial values T(0,0) = 1 and T(i,j) = 0 if j < 0 or i < j (see the Sage program below).

The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k satisfy recurrence equation p(n,x) = (1+x) * (p(n-1,x) + p'(n-1,x)) + x * p"(n-1,x) for n > 0 with initial value p(0,x) = 1 where p' and p" are first and second derivative of p. (End)

Example

Triangle begins
      1;
      1,     1;
      2,     3,      1;
      5,    12,      6,     1;
     17,    53,     39,    10,     1;
     70,   279,    260,    95,    15,    1;
    349,  1668,   1914,   880,   195,   21,   1;
   2017, 11341,  15330,  8554,  2380,  357,  28,  1;
  13358, 86019, 134317, 87626, 29379, 5530, 602, 36, 1;
Production matrix is
  1, 1;
  1, 2, 1;
  0, 4, 3,  1;
  0, 0, 9,  4,  1;
  0, 0, 0, 16,  5,  1;
  0, 0, 0,  0, 25,  6,  1;
  0, 0, 0,  0,  0, 36,  7,  1;
  0, 0, 0,  0,  0,  0, 49,  8,  1;
  0, 0, 0,  0,  0,  0,  0, 64,  9,   1;
  0, 0, 0,  0,  0,  0,  0,  0, 81,  10,  1;
  0, 0, 0,  0,  0,  0,  0,  0,  0, 100, 11, 1;

Mathematica

dim = 11; M[n_, n_] = 1; M[n_ /; 0 <= n <= dim-1, k_ /; 0 <= k <= dim-1] := M[n, k] = M[n-1, k-1] + (k+1)*M[n-1, k] + (k+1)^2*M[n-1, k+1]; M[_, _] = 0;
Table[M[n, k], {n, 0, dim-1}, {k, 0, n}] (* Jean-François Alcover, Jun 18 2019 *)

Prog

(Sage)
def A182822_triangle(dim):
    T = matrix(ZZ, dim, dim)
    for n in (0..dim-1): T[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            T[n, k] = T[n-1, k-1]+(k+1)*T[n-1, k]+(k+1)^2*T[n-1, k+1]
    return T
A182822_triangle(9) # Peter Luschny, Sep 19 2012

Crossrefs

Sequence in context: A085853 A185997 A231733 * A137211 A212275 A189036

Adjacent sequences: A182819 A182820 A182821 * A182823 A182824 A182825

Keyword

nonn,easy,tabl

Author

Paul Barry, Dec 05 2010

Status

approved