basis-map coefficients of A112227 → basis A013610 → basis A007318

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Submission text ?

Triangle begins:
   1:      16,       24,         9;
   2:      64,      144,       108,        27;
   3:     256,      768,       864,       432,        81;
   4:    1024,     3840,      5760,      4320,      1620,       243;
   5:    4096,    18432,     34560,     34560,     19440,      5832,       729;
   6:   16384,    86016,    193536,    241920,    181440,     81648,     20412,      2187;
   7:   65536,   393216,   1032192,   1548288,   1451520,    870912,    326592,     69984,     6561;
   8:  262144,  1769472,   5308416,   9289728,  10450944,   7838208,   3919104,   1259712,   236196,    19683;
   9: 1048576,  7864320,  26542080,  53084160,  69672960,  62705664,  39191040,  16796160,  4723920,   787320,   59049;
  10: 4194304, 34603008, 129761280, 291962880, 437944320, 459841536, 344881152, 184757760, 69284160, 17321040, 2598156, 177147;

The row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) for n >= 2, with P(1,x) = 16 + 24*x + 9*x^2.

P[1, x_] := 16 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 16 + 24*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = (4 + 3*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 (−∞, -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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 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 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 (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) 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)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A156992 → basis A013611 → basis A013609
• A038329 → basis A013610 → basis A013609
• A038279 → basis A013610 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A085880 → basis A013611 → basis A013609
• A038243 → basis A013609 → x ↦ x/(x+1)
• A038334 → basis A013609 → basis A013610
• A028326 → basis A013611 → basis A013609
• A013610 → basis A007318 → D
• A038335 → D
• A038329 → basis A013611 → basis A013610
• A135065 → x ↦ x+1 → basis A013611
• A178820 → basis A007318 → basis A013611
• A013609 → D → basis A013609
• A178821 → basis A007318 → basis A013611
• A013610 → basis A013611 → x ↦ x+1
• A028326 → basis A007318 → basis A013611
• A120910 → basis A013611 → D
• A038330 → basis A013610
• A027465 → D → basis A013610
• A038330 → basis A013610 → reverse
• A085880 → x ↦ x+1 → basis A013611
• A178820 → basis A013611 → basis A013609
• A120910 → basis A013611
• A038221 → basis A013611 → basis A013609
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• A038284 → D
• A038335 → reverse
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
• A356546 → basis A013611 → basis A013609
• A130123 → basis A013610 → x ↦ x+1
• A003506 → x ↦ x+1 → basis A013611
• A178822 → basis A013611 → basis A013609
• A120910 → D → basis A013611
• A178821 → basis A013611 → basis A013609
• A013610 → basis A007318
• A085881 → basis A013611 → basis A013609
• A013609 → basis A013609 → D
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• A038284 → reverse
• A094305 → x ↦ x+1 → basis A013611
• A196347 → basis A013611 → basis A013609
• A196347 → x ↦ x+1 → basis A013611
• A038330 → x ↦ x/(x+1)
• A038208 → basis A013611 → basis A013609
• A178822 → x ↦ x+1 → basis A013611
• A027465 → x ↦ x/(x+1)
• A130123 → basis A013609 → basis A013609
• A038330 → basis A013610 → D
• A013609 → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• basis-map coefficients of A026729 → basis A013610 → basis A007318
• A120909 → basis A013610 → basis A013609
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• A356546 → x ↦ x+1 → basis A013611
• A120910 → basis A013611 → reverse
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
• A136572 → basis A013610 → basis A007318

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.