A026729 → basis A013610 → D

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Triangle begins:
   1:    0;
   2:    3;
   3:    9,    18;
   4:   21,    90,     81;
   5:   45,   288,    567,    324;
   6:   90,   774,   2349,   2916,    1215;
   7:  174,  1872,   7695,  14904,   13365,    4374;
   8:  327,  4230,  21951,  57996,   81405,   56862,   15309;
   9:  603,  9108,  57105, 191484,  368145,  402408,  229635,  52488;
  10: 1095, 18918, 139077, 565704, 1385100, 2077650, 1852389, 892296, 177147;

The row polynomials satisfy (3*n^2-12*n+12-(9*n-18)*x)*P(n,x) = -(6*n^2-51*n+90-(45*n-144)*x+54*x^2)*P(n-1,x) + (5*n-15+(18*n-69)*x+(9*n-81)*x^2-27*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 - 51*n + 90 - (45*n - 144)*x + 54*x^2)*P[n - 1, x] + (5*n - 15 + (18*n - 69)*x + (9*n - 81)*x^2 - 27*x^3)*D[P[n - 1, x], x])/(3*n^2 - 12*n + 12 - (9*n - 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = (-(6*n^2 - 51*n + 90 - (45*n - 144)*x + 54*x^2)*v[i - 1] + (5*n - 15 + (18*n - 69)*x + (9*n - 81)*x^2 - 27*x^3)*deriv(v[i - 1], x))/(3*n^2 - 12*n + 12 - (9*n - 18)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 basis_map_A013610 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward ant…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • Affine, reversal and Möbius substitutions (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
        • 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 basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.?
                • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
                • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            no recorded property covers this requirement
        • ↑ Ma–Wang interval theorem (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Ma–Wang interval theorem (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Ma–Wang interval theorem (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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.