A011973 → basis A013610 → basis A007318

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Triangle begins:
  0:    1;
  1:    1;
  2:    5,    3;
  3:    9,    6;
  4:   29,   33,    9;
  5:   65,   84,   27;
  6:  181,  303,  162,   27;
  7:  441,  834,  522,  108;
  8: 1165, 2589, 2079,  702,  81;
  9: 2929, 7248, 6669, 2700, 405;

The row polynomials satisfy 3*n*P(n,x) = (27*n-24+(18*n-18)*x)*P(n-1,x) - (68+99*x+36*x^2)*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[((27*n - 24 + (18*n - 18)*x)*P[n - 1, x] - (68 + 99*x + 36*x^2)*D[P[n - 1, x], x])/(3*n)]];
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] = ((27*n - 24 + (18*n - 18)*x)*v[i - 1] - (68 + 99*x + 36*x^2)*deriv(v[i - 1], x))/(3*n)); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coef…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
        • Parent-row provers of polytool polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›

          • from delannoy?
            • 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 delannoy?
                • ↑ 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 delannoy?
                    • ↑ Liu–Wang theorem with a lag row (above)
                    • ↑ Liu–Wang theorem for three-term recurrences (above)
                    • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                      • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                        no recorded property covers this requirement
                    • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

                      • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
                        no recorded property covers this requirement
                • ↑ Affine-current basis map theorem (above)
                • ↑ Affine-current basis map theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine-current basis map theorem (above)
            • ↑ Affine-current basis map theorem (above)
        • Parent-row provers of polytool polytool ? 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Gamma criterion (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
• Ore-algebra identity of two definitions polytool ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coef…?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A052553 → basis A013610 → x ↦ x+1
• A169803 → basis A013610 → basis A007318

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