OFFSET
0,4
COMMENTS
LINKS
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
KEYWORD
AUTHOR
Paul Barry, Dec 05 2010
STATUS
approved