A008288 → basis A013610 → reverse

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Submission text ?

Triangle begins:
   1:     1;
   2:     3,      2;
   3:     9,     15,      5;
   4:    27,     72,     54,      12;
   5:    81,    297,    360,     174,      29;
   6:   243,   1134,   1917,    1476,     525,      70;
   7:   729,   4131,   8991,    9639,    5409,    1521,    169;
   8:  2187,  14580,  38880,   53784,   41850,   18432,   4284,    408;
   9:  6561,  50301, 158922,  270216,  271026,  164862,  59688,  11820,   985;
  10: 19683, 170586, 623295, 1259712, 1553742, 1215972, 605934, 186120, 32103, 2378;

The row polynomials satisfy (9*n-9)*P(n,x) = (27*n-27+(27*n^2-63*n+36)*x+(24*n^2-108*n+120)*x^2+(8*n^2-40*n+48)*x^3)*P(n-1,x) - ((27*n-27)*x+(90*n-207)*x^2+(72*n-204)*x^3+(16*n-48)*x^4)*d/dx P(n-1,x) + (27*x^2+81*x^3+48*x^4+8*x^5)*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[((27*n - 27 + (27*n^2 - 63*n + 36)*x + (24*n^2 - 108*n + 120)*x^2 + (8*n^2 - 40*n + 48)*x^3)*P[n - 1, x] - ((27*n - 27)*x + (90*n - 207)*x^2 + (72*n - 204)*x^3 + (16*n - 48)*x^4)*D[P[n - 1, x], x] + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*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] = ((27*n - 27 + (27*n^2 - 63*n + 36)*x + (24*n^2 - 108*n + 120)*x^2 + (8*n^2 - 40*n + 48)*x^3)*v[i - 1] - ((27*n - 27)*x + (90*n - 207)*x^2 + (72*n - 204)*x^3 + (16*n - 48)*x^4)*deriv(v[i - 1], x) + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*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 (−∞, 0]
  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 ?

Same fingerprint ?

• delannoy → basis A013610 → reverse
• A008288 → basis A013610
• delannoy → basis A013610

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