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

%I A160486 #7 Jul 22 2025 06:25:01
%S A160486 1,1,1,1,18,5,1,179,479,61,1,1636,18270,19028,1385,1,14757,540242,
%T A160486 1949762,1073517,50521,1,132854,14494859,137963364,241595239,82112518,
%U A160486 2702765
%N A160486 Triangle of polynomial coefficients related to the o.g.f.s. of the RBS1 polynomials.
%C A160486 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 A160486 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 A160486 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 A160486 The Maple program given below is derived from the one given in A008971.
%e A160486 The first few rows of the triangle are:
%e A160486 [1]
%e A160486 [1, 1]
%e A160486 [1, 18, 5]
%e A160486 [1, 179, 479, 61]
%e A160486 [1, 1636, 18270, 19028, 1385]
%e A160486 The first few RB(z,1-2*m) polynomials are:
%e A160486 RB(z,-1) = 1
%e A160486 RB(z,-3) = z+1
%e A160486 RB(z,-5) = z^2+18*z+5
%e A160486 RB(z,-7) = z^3+179*z^2+479*z+61
%e A160486 The first few GFRBS1(z,1-2*m) are:
%e A160486 GFRBS1(z,-1) = (-1)*(1)/(z-1)
%e A160486 GFRBS1(z,-3) = (-1)*(z+1)/(z-1)^2
%e A160486 GFRBS1(z,-5) = (-1)*(z^2+18*z+5)/(z-1)^3
%e A160486 GFRBS1(z,-7) = (-1)*(z^3+179*z^2+479*z+61)/(z-1)^4
%p A160486 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 A160486 Cf. A160480 and A160485.
%Y A160486 The row sums equal A010050.
%Y A160486 This triangle is a sub-triangle of A008971.
%Y A160486 A000340(2*n-2), A000363(2*n+2) and A000507(2*n+4) equal the second, third and fourth left hand columns.
%Y A160486 The first right hand column equals the Euler numbers A000364.
%K A160486 easy,nonn,tabl
%O A160486 1,5
%A A160486 _Johannes W. Meijer_, May 24 2009, Sep 19 2012