OFFSET
1,5
COMMENTS
As we showed in A160485 the n-th term of the coefficients of matrix row BS1[1-2*m,n] for m = 1 , 2, 3, .. , can be generated with the RBS1(1-2*m,n) polynomials.
We define the o.g.f.s. of these polynomials by GFRBS1(z,1-2*m) = sum(RBS1(1-2*m,n)*z^(n-1), n=1..infinity) for m = 1, 2, 3, .. . The general expression of these o.g.f.s. is GFRBS1(z,1-2*m) = (-1)*RB(z,1-2*m)/(z-1)^m.
The RB(z,1-2*m) polynomials lead to a triangle that is a subtriangle of the 'double triangle' A008971. The even rows of the latter triangle are identical to the rows of our triangle.
The Maple program given below is derived from the one given in A008971.
EXAMPLE
The first few rows of the triangle are:
[1]
[1, 1]
[1, 18, 5]
[1, 179, 479, 61]
[1, 1636, 18270, 19028, 1385]
The first few RB(z,1-2*m) polynomials are:
RB(z,-1) = 1
RB(z,-3) = z+1
RB(z,-5) = z^2+18*z+5
RB(z,-7) = z^3+179*z^2+479*z+61
The first few GFRBS1(z,1-2*m) are:
GFRBS1(z,-1) = (-1)*(1)/(z-1)
GFRBS1(z,-3) = (-1)*(z+1)/(z-1)^2
GFRBS1(z,-5) = (-1)*(z^2+18*z+5)/(z-1)^3
GFRBS1(z,-7) = (-1)*(z^3+179*z^2+479*z+61)/(z-1)^4
MAPLE
nmax:=15; G := sqrt(1-t)/(sqrt(1-t)*cosh(x*sqrt(1-t))-sinh(x*sqrt(1-t))): Gser := simplify(series(G, x=0, nmax+1)): for m from 0 to nmax do P[m] := sort(expand(m!* coeff(Gser, x, m))) od: nmx := floor(nmax/2); for n from 0 to nmx do for k from 0 to nmx-1 do A(n+1, n+1-k) := coeff(P[2*n], t, n-k) od: od: seq(seq(A(n, m), m=1..n), n=1..nmx);
CROSSREFS
KEYWORD
AUTHOR
Johannes W. Meijer, May 24 2009, Sep 19 2012
STATUS
approved