basis-map coefficients of A112227 → basis A013609 → basis A013611

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Submission text ?

Triangle begins:
   1:      9,      48,       64;
   2:     27,     216,      576,       512;
   3:     81,     864,     3456,      6144,       4096;
   4:    243,    3240,    17280,     46080,      61440,       32768;
   5:    729,   11664,    77760,    276480,     552960,      589824,      262144;
   6:   2187,   40824,   326592,   1451520,    3870720,     6193152,     5505024,     2097152;
   7:   6561,  139968,  1306368,   6967296,   23224320,    49545216,    66060288,    50331648,    16777216;
   8:  19683,  472392,  5038848,  31352832,  125411328,   334430208,   594542592,   679477248,   452984832,   134217728;
   9:  59049, 1574640, 18895680, 134369280,  627056640,  2006581248,  4459069440,  6794772480,  6794772480,  4026531840,  1073741824;
  10: 177147, 5196312, 69284160, 554273280, 2956124160, 11036196864, 29429858304, 56056872960, 74742497280, 66437775360, 35433480192, 8589934592;

The row polynomials satisfy P(n,x) = (3+8*x)*P(n-1,x) for n >= 2, with P(1,x) = 9 + 48*x + 64*x^2.

P[1, x_] := 9 + 48*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 9 + 48*x + 64*x^2; for(n = 2, N, my(i = n); v[i] = (3 + 8*x)*v[i - 1]); 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • Parent-row provers of polytool polytool ? step › direct check
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (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 A scaled Hermite triangle.?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of A scaled Hermite triangle.?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → basis A013609 → basis A013611
• A038328 → x ↦ x+1 → basis A013610
• A038329 → basis A013610 → basis A007318
• A130123 → basis A013609 → basis A013611
• A038327 → basis A013610 → basis A013609
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• A120909 → basis A013610 → basis A013611
• A027466 → basis A013610
• A013609 → D → basis A013611
• A178820 → basis A013611 → basis A013611
• A013622 → x ↦ x/(x+1) → D
• A038226 → D
• A038208 → basis A013611 → basis A013611
• A027466 → basis A013610 → D
• A136572 → basis A013609 → basis A013611
• A178821 → basis A013611 → basis A013611
• A120909 → basis A013611
• A013609 → basis A013610 → basis A013611
• A013612 → basis A013609 → x ↦ x/(x+1)
• A038281 → D
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• A028326 → basis A013611 → basis A013611
• A013622 → x ↦ x/(x+1)
• A038271 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013611

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.