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Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials
4

%I #5 Sep 19 2012 11:24:31

%S 1,1,1,1,18,5,1,179,479,61,1,1636,18270,19028,1385,1,14757,540242,

%T 1949762,1073517,50521,1,132854,14494859,137963364,241595239,82112518,

%U 2702765

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

%C 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.

%C 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.

%C 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.

%C The Maple program given below is derived from the one given in A008971.

%e The first few rows of the triangle are:

%e [1]

%e [1, 1]

%e [1, 18, 5]

%e [1, 179, 479, 61]

%e [1, 1636, 18270, 19028, 1385]

%e The first few RB(z,1-2*m) polynomials are:

%e RB(z,-1) = 1

%e RB(z,-3) = z+1

%e RB(z,-5) = z^2+18*z+5

%e RB(z,-7) = z^3+179*z^2+479*z+61

%e The first few GFRBS1(z,1-2*m) are:

%e GFRBS1(z,-1) = (-1)*(1)/(z-1)

%e GFRBS1(z,-3) = (-1)*(z+1)/(z-1)^2

%e GFRBS1(z,-5) = (-1)*(z^2+18*z+5)/(z-1)^3

%e GFRBS1(z,-7) = (-1)*(z^3+179*z^2+479*z+61)/(z-1)^4

%p 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);

%Y Cf. A160480 and A160485.

%Y The row sums equal A010050.

%Y This triangle is a sub-triangle of A008971.

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

%Y The first right hand column equals the Euler numbers A000364.

%K easy,nonn,tabl

%O 1,5

%A _Johannes W. Meijer_, May 24 2009, Sep 19 2012