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A160468 Triangle of polynomial coefficients related to the o.g.f.s of the RES1 polynomials.

%I A160468 #16 Apr 22 2020 22:29:58
%S A160468 1,1,2,1,17,26,2,62,192,60,1,1382,7192,5097,502,2,21844,171511,217186,
%T A160468 55196,2036,2,929569,10262046,20376780,9893440,1089330,16356,4,
%U A160468 6404582,94582204,271154544,215114420,48673180,2567568,16376,1
%N A160468 Triangle of polynomial coefficients related to the o.g.f.s of the RES1 polynomials.
%C A160468 In A160464 we defined the ES1 matrix by ES1[2*m-1,n=1] and in A094665 it was shown that the n-th term of the coefficients of matrix row ES1[1-2*m,n] for m >= 1 can be generated with the RES1(1-2*m,n) polynomials.
%C A160468 We define the o.g.f.s. of these polynomials by GFRES1(z,1-2*m) = sum(RES1(1-2*m,n)*z^(n-1), n=1..infinity) for m >= 1. The general expression of the o.g.f.s. is GFRES1(z,1-2*m) = (-1)*RE(z,1-2*m)/(2*p(m-1)*(z-1)^(m)). The p(m-1), m >= 1, sequence is Gould's sequence A001316.
%C A160468 The coefficients of the RE(z,1-2*m) polynomials lead to the triangle given above.
%C A160468 The E(z,n) = numer(sum((-1)^(n+1)*k^n*z^(k-1), k=1..infinity)) polynomials with n >= 1, see the Maple algorithm, lead to the Eulerian numbers A008292.
%C A160468 Some of our results are conjectures based on numerical evidence.
%H A160468 Grzegorz Rzadkowski, M Urlinska, <a href="http://arxiv.org/abs/1612.06635">A Generalization of the Eulerian Numbers</a>, arXiv preprint arXiv:1612.06635, 2016
%e A160468 The first few rows are:
%e A160468 [1]
%e A160468 [1]
%e A160468 [2, 1]
%e A160468 [17, 26, 2]
%e A160468 [62, 192, 60, 1]
%e A160468 The first few polynomials RE(z,m) are:
%e A160468 RE(z,-1) = 1
%e A160468 RE(z,-3) = 1
%e A160468 RE(z,-5) = 2+z
%e A160468 RE(z,-7) = 17+26*z+2*z^2
%e A160468 The first few GFRES1(z,m) are:
%e A160468 GFRES1(z,-1) = -(1/1)*(1)/(2*(z-1)^1)
%e A160468 GFRES1(z,-3) = -(1/2)*(1)/(2*(z-1)^2)
%e A160468 GFRES1(z,-5) = -(1/2)*(2+z)/(2*(z-1)^3)
%e A160468 GFRES1(z,-7) = -(1/4)*(17+26*z+2*z^2)/(2*(z-1)^4)
%p A160468 nmax := 8; mmax := nmax: T(0, x) := 1: for i from 1 to nmax do dgr := degree(T(i-1, x), x): for na from 0 to dgr do c(na) := coeff(T(i-1, x), x, na) od: T(i-1, x+1) := 0: for nb from 0 to dgr do T(i-1, x+1) := T(i-1, x+1) + c(nb)*(x+1)^nb od: for nc from 0 to dgr do ECGP(i-1, nc+1) := coeff(T(i-1, x), x, nc) od: T(i, x) := expand((2*x+1)*(x+1)*T(i-1, x+1) - 2*x^2*T(i-1, x)) od: dgr := degree (T(nmax, x), x): kmax := nmax: for k from 1 to kmax do p := k: for m from 1 to k do E(m, k) := sum((-1)^(m-q)*(q^k)*binomial(k+1, m-q), q=1..m) od: fx(p) := (-1)^(p+1) * (sum(E(r, k)*z^(k-r), r=1..k))/(z-1)^(p+1): GF(-(2*p+1)) := sort(simplify(((-1)^p* 1/2^(p+1)) * sum(ECGP(k-1, k-s)*fx(k-s), s=0..k-1)), ascending): NUMGF(-(2*p+1)) := -numer(GF(-(2*p+1))): for n from 1 to mmax+1 do A(k+1, n) := coeff(NUMGF(-(2*p+1)), z, n-1) od: od: for m from 2 to mmax do A(1, m) := 0 od: A(1, 1) := 1: FT(1) := 1: for n from 1 to nmax do for m from 1 to n do FT((n)*(n-1)/2+m+1) := A(n+1, m) end do end do: a := n-> FT(n): seq(a(n), n = 1..(nmax+1)*(nmax)/2+1);
%t A160468 T[ n_, k_] := Coefficient[a[2 n]/2^IntegerExponent[(2 n)!, 2], x, n + k];
%t A160468 a[0] = a[1] = 1; a[ m_] := a[m] = With[{n = m - 1}, x Sum[ a[k] a[n - k] Binomial[n, k], {k, 0, n}]]; Join[{1}, Flatten@Table[T[n, k], {n, 1, 8}, {k, 0, n - 1}]] (* _Michael Somos_, Apr 22 2020 *)
%Y A160468 Cf. A160464, A094665 and A083061.
%Y A160468 For the Eulerian numbers E(n, k) see A008292.
%Y A160468 The p(n) sequence equals Gould's sequence A001316.
%Y A160468 The first right hand column of the triangle equals A048896.
%Y A160468 The first left hand column equals A160469.
%Y A160468 The row sums equal the absolute values of A117972.
%K A160468 easy,nonn,tabf
%O A160468 1,3
%A A160468 _Johannes W. Meijer_, May 24 2009
%E A160468 Edited by _Johannes W. Meijer_, Sep 23 2012