OFFSET
0,4
COMMENTS
The table entries are defined by a recurrence relation (see below).
This triangle can be used to calculate the entries of A186491: the nonzero entries of the first column of the triangle give A186491.
PRODUCTION MATRIX
The production matrix P for this triangle is the bidiagonal matrix with the sequence [2,4,6,...] on the main subdiagonal, the sequence [1,3,5,...] on the main superdiagonal and 0's elsewhere: the first row of P^n is the n-th row of this triangle.
LINKS
C. V. Sukumar and A. Hodges, Quantum algebras and parity-dependent spectra, Proc. R. Soc. A (2007) 463, 2415-2427.
FORMULA
Recurrence relation
(1)... T(n,k) = (2*k-1)*T(n-1,k-1)+(2*k+2)*T(n-1,k+1).
GENERATING FUNCTION
E.g.f. (Compare with the e.g.f. of A104035):
(2)... 1/sqrt(cos(2*t)-u*sin(2*t)) = sum {n = 0..inf } R(n,u)*t^n/n! = 1 + u*t + (2+3*u^2)*t^2/2! + (14*u+15*u^3)*t^3/3!+....
ROW POLYNOMIALS
The row polynomials R(n,u) begin
... R(1,u) = u
... R(2,u) = 2+3*u^2
... R(3,u) = 14*u+15*u^3
... R(4,u) = 28+132*u^2+105u^4.
They satisfy the recurrence relation
(3)... R(n+1,u) = 2*(1+u^2)*d/du(R(n,u))+u*R(n,u) with starting value R(0,u) = 1.
Compare with Formula (1) of A104035 for the polynomials Q_n(u).
The polynomials R(n,u) are related to the shifted row polynomials A(n,u) of A142459 via
(4)... R(n,u) = ((u+I)/2)^n*A(n+1,(u-I)/(u+I))
with the inverse identity
(5)... A(n+1,u) = (-I)^n*(1-u)^n*R(n,I*(1+u)/(1-u)),
where {A(n,u)}n>=1 begins [1,1+u,1+10*u+u^2,1+59*u+59*u^2+u^3,...] and I = sqrt(-1).
EXAMPLE
Table begins
n\k|.....0.....1......2.....3......4.....5......6
=================================================
0..|.....1
1..|.....0.....1
2..|.....2.....0......3
3..|.....0....14......0....15
4..|....28.....0....132.....0....105
5..|.....0...556......0..1500......0...945
6..|..1112.....0..10668.....0..19950.....0..10395
..
Examples of recurrence relation
T(4,2) = 3*T(3,1) + 6*T(3,3) = 3*14 + 6*15 = 132;
T(6,4) = 7*T(5,3) + 10*T(5,5) = 7*1500 + 10*945 = 19950.
MATHEMATICA
R[0][_] = 1; R[1][u_] = u;
R[n_][u_] := R[n][u] = 2(1+u^2) R[n-1]'[u] + u R[n-1][u];
Table[CoefficientList[R[n][u], u], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 13 2019 *)
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Feb 22 2011
STATUS
approved