A008288 → basis A013610 → x ↦ x+1

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Submission text ?

Triangle begins:
   1:       1;
   2:       5,        3;
   3:      29,       33,        9;
   4:     165,      279,      153,       27;
   5:     941,     2109,     1737,      621,       81;
   6:    5365,    14979,    16461,     8883,     2349,      243;
   7:   30589,   102249,   140517,   101493,    40581,     8505,      729;
   8:  174405,   679023,  1120113,  1013931,   543429,   172287,    29889,    2187;
   9:  994381,  4419093,  8506449,  9257517,  6225741,  2647485,   694737,  102789,   6561;
  10: 5669525, 28317915, 62307045, 79222995, 64116765, 34234083, 12052557, 2696571, 347733, 19683;

The row polynomials satisfy (9*n-9)*P(n,x) = (18*n^2-108*n+189-(27*n-81)*x)*P(n-1,x) - (123*n-498+(126*n-549)*x+(27*n-135)*x^2)*d/dx P(n-1,x) + (164+291*x+162*x^2+27*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 108*n + 189 - (27*n - 81)*x)*P[n - 1, x] - (123*n - 498 + (126*n - 549)*x + (27*n - 135)*x^2)*D[P[n - 1, x], x] + (164 + 291*x + 162*x^2 + 27*x^3)*D[P[n - 1, x], {x, 2}])/(9*n - 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((18*n^2 - 108*n + 189 - (27*n - 81)*x)*v[i - 1] - (123*n - 498 + (126*n - 549)*x + (27*n - 135)*x^2)*deriv(v[i - 1], x) + (164 + 291*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 1], x), x))/(9*n - 9)); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) 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 ?

Candidate equalities ?

• A008288 → basis A013610 → basis A007318 15 rows; no all-index equality certificate was found

Same fingerprint ?

• delannoy → basis A013610 → basis A007318
• delannoy → basis A013610 → x ↦ x+1

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