A008288 → basis A013610 → D

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Submission text ?

Triangle begins:
   1:     0;
   2:     3;
   3:    15,     18;
   4:    54,    144,      81;
   5:   174,    720,     891,     324;
   6:   525,   2952,    5751,    4536,    1215;
   7:  1521,  10818,   28917,   35964,   20655,    4374;
   8:  4284,  36864,  125550,  215136,  194400,   87480,   15309;
   9: 11820, 119376,  494586, 1084104, 1351080,  953532,  352107,   52488;
  10: 32103, 372240, 1817802, 4863888, 7768710, 7558272, 4363065, 1364688, 177147;

The row polynomials satisfy (12*n-24+(18*n^2-54*n+36)*x-(27*n-54)*x^2)*P(n,x) = -(24*n^2-156*n+192+(54*n^2-486*n+720)*x-(243*n-648)*x^2+162*x^3)*P(n-1,x) + (16*n-32+(96*n-216)*x+(162*n-468)*x^2+(54*n-351)*x^3-81*x^4)*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[(-(24*n^2 - 156*n + 192 + (54*n^2 - 486*n + 720)*x - (243*n - 648)*x^2 + 162*x^3)*P[n - 1, x] + (16*n - 32 + (96*n - 216)*x + (162*n - 468)*x^2 + (54*n - 351)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(12*n - 24 + (18*n^2 - 54*n + 36)*x - (27*n - 54)*x^2)]];
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] = (-(24*n^2 - 156*n + 192 + (54*n^2 - 486*n + 720)*x - (243*n - 648)*x^2 + 162*x^3)*v[i - 1] + (16*n - 32 + (96*n - 216)*x + (162*n - 468)*x^2 + (54*n - 351)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(12*n - 24 + (18*n^2 - 54*n + 36)*x - (27*n - 54)*x^2)); 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 Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
        • 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 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
            • ↑ 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 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
                • ↑ 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 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
                • 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
            • ↑ 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)
    • Geometric basis map (elementary) polytool ? step ›

      • from Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.?
        • ↑ Liu–Wang theorem with a lag row (above)
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine-current basis map theorem (above)
        • ↑ Affine-current basis map theorem (above)

Related ?

Same fingerprint ?

• delannoy → basis A013610 → D

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.