A081580 → γ → D+1

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Triangle begins:
  0:  1;
  1:  1;
  2:  6,    5;
  3: 11,   10;
  4: 16,   65,   25;
  5: 21,  170,   75;
  6: 26,  325,  525,   125;
  7: 31,  530, 1750,   500;
  8: 36,  785, 4125,  3750,  625;
  9: 41, 1090, 8025, 15000, 3125;

The row polynomials satisfy (5*n^3-14*n^2+9*n+(20*n^2-49*n)*x+20*n*x^2)*P(n,x) = (5*n^3-9*n^2+4*n+(50*n^3-170*n^2+91*n)*x+(200*n^2-470*n+290)*x^2+(200*n-200)*x^3)*P(n-1,x) - ((5*n^2-4*n)*x+(100*n^2-60*n-9)*x^2+(400*n-160)*x^3+400*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^3 - 9*n^2 + 4*n + (50*n^3 - 170*n^2 + 91*n)*x + (200*n^2 - 470*n + 290)*x^2 + (200*n - 200)*x^3)*P[n - 1, x] - ((5*n^2 - 4*n)*x + (100*n^2 - 60*n - 9)*x^2 + (400*n - 160)*x^3 + 400*x^4)*D[P[n - 1, x], x])/(5*n^3 - 14*n^2 + 9*n + (20*n^2 - 49*n)*x + 20*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^3 - 9*n^2 + 4*n + (50*n^3 - 170*n^2 + 91*n)*x + (200*n^2 - 470*n + 290)*x^2 + (200*n - 200)*x^3)*v[i - 1] - ((5*n^2 - 4*n)*x + (100*n^2 - 60*n - 9)*x^2 + (400*n - 160)*x^3 + 400*x^4)*deriv(v[i - 1], x))/(5*n^3 - 14*n^2 + 9*n + (20*n^2 - 49*n)*x + 20*n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from gamma_polynomial of Pascal-(1,5,1) array.?
    • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

      • from Pascal-(1,5,1) array.?
        • Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Pascal-(1,5,1) array.?
            • ↑ Liu–Wang theorem with a lag row (above)
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Pascal-(1,5,1) array.?
                • ↑ Liu–Wang theorem with a lag row (above)
                • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.