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Exponential Riordan array, defining orthogonal polynomials related to permutations without double falls.
1

%I #35 Mar 28 2022 10:55:49

%S 1,1,1,2,3,1,5,12,6,1,17,53,39,10,1,70,279,260,95,15,1,349,1668,1914,

%T 880,195,21,1,2017,11341,15330,8554,2380,357,28,1,13358,86019,134317,

%U 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

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

%C 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.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry2/barry281.html">Constructing Exponential Riordan Arrays from Their A and Z Sequences</a>, Journal of Integer Sequences, 17 (2014), #14.2.6.

%F 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))].

%F From _Werner Schulte_, Mar 27 2022: (Start)

%F 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).

%F 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)

%e Triangle begins

%e 1;

%e 1, 1;

%e 2, 3, 1;

%e 5, 12, 6, 1;

%e 17, 53, 39, 10, 1;

%e 70, 279, 260, 95, 15, 1;

%e 349, 1668, 1914, 880, 195, 21, 1;

%e 2017, 11341, 15330, 8554, 2380, 357, 28, 1;

%e 13358, 86019, 134317, 87626, 29379, 5530, 602, 36, 1;

%e Production matrix is

%e 1, 1;

%e 1, 2, 1;

%e 0, 4, 3, 1;

%e 0, 0, 9, 4, 1;

%e 0, 0, 0, 16, 5, 1;

%e 0, 0, 0, 0, 25, 6, 1;

%e 0, 0, 0, 0, 0, 36, 7, 1;

%e 0, 0, 0, 0, 0, 0, 49, 8, 1;

%e 0, 0, 0, 0, 0, 0, 0, 64, 9, 1;

%e 0, 0, 0, 0, 0, 0, 0, 0, 81, 10, 1;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 11, 1;

%t 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;

%t Table[M[n, k], {n, 0, dim-1}, {k, 0, n}] (* _Jean-François Alcover_, Jun 18 2019 *)

%o (Sage)

%o def A182822_triangle(dim):

%o T = matrix(ZZ,dim,dim)

%o for n in (0..dim-1): T[n,n] = 1

%o for n in (1..dim-1):

%o for k in (0..n-1):

%o T[n,k] = T[n-1,k-1]+(k+1)*T[n-1,k]+(k+1)^2*T[n-1,k+1]

%o return T

%o A182822_triangle(9) # _Peter Luschny_, Sep 19 2012

%K nonn,easy,tabl

%O 0,4

%A _Paul Barry_, Dec 05 2010