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   A160468 Triangle of polynomial coefficients related to the o.g.f.s of the RES1 polynomials. 7
   1, 1, 2, 1, 17, 26, 2, 62, 192, 60, 1, 1382, 7192, 5097, 502, 2, 21844, 171511, 217186, 55196, 2036, 2, 929569, 10262046, 20376780, 9893440, 1089330, 16356, 4, 6404582, 94582204, 271154544, 215114420, 48673180, 2567568, 16376, 1 (list; graph; refs; listen; history; text; internal format)
   OFFSET 1,3 COMMENTS 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. 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. The coefficients of the RE(z,1-2*m) polynomials lead to the triangle given above. 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. Some of our results are conjectures based on numerical evidence. LINKS Grzegorz Rzadkowski, M Urlinska, A Generalization of the Eulerian Numbers, arXiv preprint arXiv:1612.06635, 2016 EXAMPLE The first few rows are: [1] [1] [2, 1] [17, 26, 2] [62, 192, 60, 1] The first few polynomials RE(z,m) are: RE(z,-1) = 1 RE(z,-3) = 1 RE(z,-5) = 2+z RE(z,-7) = 17+26*z+2*z^2 The first few GFRES1(z,m) are: GFRES1(z,-1) = -(1/1)*(1)/(2*(z-1)^1) GFRES1(z,-3) = -(1/2)*(1)/(2*(z-1)^2) GFRES1(z,-5) = -(1/2)*(2+z)/(2*(z-1)^3) GFRES1(z,-7) = -(1/4)*(17+26*z+2*z^2)/(2*(z-1)^4) MAPLE 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); MATHEMATICA T[ n_, k_] := Coefficient[a[2 n]/2^IntegerExponent[(2 n)!, 2], x, n + k]; 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 *) CROSSREFS Cf. A160464, A094665 and A083061. For the Eulerian numbers E(n, k) see A008292. The p(n) sequence equals Gould's sequence A001316. The first right hand column of the triangle equals A048896. The first left hand column equals A160469. The row sums equal the absolute values of A117972. Sequence in context: A266827 A316226 A352138 * A242195 A012889 A013072 Adjacent sequences: A160465 A160466 A160467 * A160469 A160470 A160471 KEYWORD easy,nonn,tabf AUTHOR Johannes W. Meijer, May 24 2009 EXTENSIONS Edited by Johannes W. Meijer, Sep 23 2012 STATUS approved
  
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  Last modified February 3 14:36 EST 2023. Contains 360035 sequences. (Running on oeis4.)
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