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

%I A182822 #35 Mar 28 2022 10:55:49
%S A182822 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 A182822 880,195,21,1,2017,11341,15330,8554,2380,357,28,1,13358,86019,134317,
%U A182822 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 A182822 Exponential Riordan array, defining orthogonal polynomials related to permutations without double falls.
%C A182822 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 A182822 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 A182822 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 A182822 From _Werner Schulte_, Mar 27 2022: (Start)
%F A182822 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 A182822 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 A182822 Triangle begins
%e A182822       1;
%e A182822       1,     1;
%e A182822       2,     3,      1;
%e A182822       5,    12,      6,     1;
%e A182822      17,    53,     39,    10,     1;
%e A182822      70,   279,    260,    95,    15,    1;
%e A182822     349,  1668,   1914,   880,   195,   21,   1;
%e A182822    2017, 11341,  15330,  8554,  2380,  357,  28,  1;
%e A182822   13358, 86019, 134317, 87626, 29379, 5530, 602, 36, 1;
%e A182822 Production matrix is
%e A182822   1, 1;
%e A182822   1, 2, 1;
%e A182822   0, 4, 3,  1;
%e A182822   0, 0, 9,  4,  1;
%e A182822   0, 0, 0, 16,  5,  1;
%e A182822   0, 0, 0,  0, 25,  6,  1;
%e A182822   0, 0, 0,  0,  0, 36,  7,  1;
%e A182822   0, 0, 0,  0,  0,  0, 49,  8,  1;
%e A182822   0, 0, 0,  0,  0,  0,  0, 64,  9,   1;
%e A182822   0, 0, 0,  0,  0,  0,  0,  0, 81,  10,  1;
%e A182822   0, 0, 0,  0,  0,  0,  0,  0,  0, 100, 11, 1;
%t A182822 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 A182822 Table[M[n, k], {n, 0, dim-1}, {k, 0, n}] (* _Jean-François Alcover_, Jun 18 2019 *)
%o A182822 (Sage)
%o A182822 def A182822_triangle(dim):
%o A182822     T = matrix(ZZ,dim,dim)
%o A182822     for n in (0..dim-1): T[n,n] = 1
%o A182822     for n in (1..dim-1):
%o A182822         for k in (0..n-1):
%o A182822             T[n,k] = T[n-1,k-1]+(k+1)*T[n-1,k]+(k+1)^2*T[n-1,k+1]
%o A182822     return T
%o A182822 A182822_triangle(9) # _Peter Luschny_, Sep 19 2012
%K A182822 nonn,easy,tabl
%O A182822 0,4
%A A182822 _Paul Barry_, Dec 05 2010