basis-map coefficients of A026729, shifted → basis A013609 → basis A008277

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Triangle begins:
   1:      1;
   2:     -3,       -2;
   3:      9,       16,         4;
   4:    -27,      -98,       -60,        -8;
   5:     81,      544,       616,       192,        16;
   6:   -243,    -2882,     -5400,     -2960,      -560,       -32;
   7:    729,    14896,     43564,     37440,     12080,      1536,       64;
   8:  -2187,   -75938,   -334740,   -424088,   -207760,    -44128,    -4032,    -128;
   9:   6561,   384064,   2495056,   4486272,   3133536,    989184,   148736,   10240,    256;
  10: -19683, -1933442, -18233520, -45366560, -43441440, -19126464, -4209408, -471552, -25344, -512;

The row polynomials satisfy P(n,x) = -(3+2*x)*P(n-1,x) - 2*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(3 + 2*x)*P[n - 1, x] - 2*x*D[P[n - 1, x], x]];
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] = -(3 + 2*x)*v[i - 1] - 2*x*deriv(v[i - 1], 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
    • 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…?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (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…?
            • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
        • Affine, reversal and Möbius substitutions (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…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • 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…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A013609 → basis A013610 → basis A008277
• A028326 → basis A013611 → basis A008277
• A038208 → basis A013611 → basis A008277
• A156992 → basis A013611 → basis A008277
• A178820 → basis A013611 → basis A008277
• A196347 → basis A013611 → basis A008277
• basis-map coefficients of A026729 → basis A013609 → basis A008277
• A120909 → basis A008277
• A178822 → basis A013611 → basis A008277
• A013609 → D → basis A008277
• A085880 → basis A013611 → basis A008277
• A038221 → basis A013611 → basis A008277
• basis-map coefficients of A030528, shifted → basis A013609 → basis A008277
• A120909 → basis A013610 → basis A008277
• A178821 → basis A013611 → basis A008277
• basis-map coefficients of A112227 → basis A013609 → basis A008277
• A013609 → basis A008277
• A130123 → basis A013609 → basis A008277
• A085881 → basis A013611 → basis A008277
• A136572 → basis A013609 → basis A008277
• A356546 → basis A013611 → basis A008277

References

1. [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