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A182822 Exponential Riordan array, defining orthogonal polynomials related to permutations without double falls. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)