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

Author

Paul Barry, Dec 05 2010

Keywords

• nonn
• easy
• tabl

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.

Examples

			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;
		

Links

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

Programs

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 *)

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

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)