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).
