The row polynomials SK(n,x) are the umbral compositional inverses of the polynomials SKv(n,x) of A119467; i.e., umbrally SKv(n,SK(.,x)) = x^n = SK(n,SKv(.,x)). Therefore, this entry's matrix and A119467 are an inverse pair. Both sequences of polynomials are Appell sequences; i.e., d/dx P(n,x) = n * P(n-1,x) and (P(.,x)+y)^n = P(n,x+y). In particular, (SK(.,0)+x)^n = SK(n,x), reflecting that the first column has the e.g.f. sech(t). The raising operator is R = x - tanh(d/dx); i.e., R SK(n,x) = SK(n+1,x). The coefficients of this operator are basically the signed and aerated zag numbers A000182, which can be expressed as normalized Bernoulli numbers. The triangle is formed by multiplying the n-th diagonal of the lower triangular Pascal matrix by the Taylor series coefficient a(n) of sech(x). More relations for this type of triangle and its inverse are given by the formalism of A133314. - Tom Copeland, Sep 05 2015
The unsigned array is generated by the raising operator RTN = x + tan(d/dx) and has e.g.f. sec(t) e^(tx). The raising operator for the Appell sequence with the base sequence of polynomials A046802, enumerating the number of cells of the positive grassmannians, specializes to R above when t = -1 in the raising operator of A046802. The e.g.f. of this entry satisfies the differential equation df(x,t)/dt = R f(x,t), and the e.g.f. for the unsigned array satisfies df(x,t)/dt = RTN f(x,t). - Tom Copeland, Oct 14 2015