A160486 Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials.

1, 1, 1, 1, 18, 5, 1, 179, 479, 61, 1, 1636, 18270, 19028, 1385, 1, 14757, 540242, 1949762, 1073517, 50521, 1, 132854, 14494859, 137963364, 241595239, 82112518, 2702765

Offset: 1

Author

Johannes W. Meijer, May 24 2009, Sep 19 2012

Keywords

• easy
• nonn
• tabl

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.

Examples

			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
		

Crossrefs

Cf. A160480 and A160485.

The row sums equal A010050.

This triangle is a sub-triangle of A008971.

A000340(2*n-2), A000363(2*n+2) and A000507(2*n+4) equal the second, third and fourth left hand columns.

The first right hand column equals the Euler numbers A000364.

Programs

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);