A008288 → basis A013611 → reverse

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Submission text ?

Triangle begins:
   1:      1;
   2:      4,       2;
   3:     16,      20,       5;
   4:     64,     128,      72,      12;
   5:    256,     704,     640,     232,      29;
   6:   1024,    3584,    4544,    2624,     700,      70;
   7:   4096,   17408,   28416,   22848,    9616,    2028,     169;
   8:  16384,   81920,  163840,  169984,   99200,   32768,    5712,    408;
   9:  65536,  376832,  892928, 1138688,  856576,  390784,  106112,  15760,   985;
  10: 262144, 1703936, 4669440, 7077888, 6547456, 3843072, 1436288, 330880, 42804, 2378;

The row polynomials satisfy (2*n-2)*P(n,x) = (8*n-8+(6*n^2-14*n+8)*x+(4*n^2-18*n+20)*x^2+(n^2-5*n+6)*x^3)*P(n-1,x) - ((8*n-8)*x+(20*n-46)*x^2+(12*n-34)*x^3+(2*n-6)*x^4)*d/dx P(n-1,x) + (8*x^2+18*x^3+8*x^4+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[((8*n - 8 + (6*n^2 - 14*n + 8)*x + (4*n^2 - 18*n + 20)*x^2 + (n^2 - 5*n + 6)*x^3)*P[n - 1, x] - ((8*n - 8)*x + (20*n - 46)*x^2 + (12*n - 34)*x^3 + (2*n - 6)*x^4)*D[P[n - 1, x], x] + (8*x^2 + 18*x^3 + 8*x^4 + x^5)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
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] = ((8*n - 8 + (6*n^2 - 14*n + 8)*x + (4*n^2 - 18*n + 20)*x^2 + (n^2 - 5*n + 6)*x^3)*v[i - 1] - ((8*n - 8)*x + (20*n - 46)*x^2 + (12*n - 34)*x^3 + (2*n - 6)*x^4)*deriv(v[i - 1], x) + (8*x^2 + 18*x^3 + 8*x^4 + x^5)*deriv(deriv(v[i - 1], x), x))/(2*n - 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 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_A013611 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 A013611 → reverse
• A008288 → basis A013611
• delannoy → basis A013611

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