basis-map coefficients of A112227 → basis A013609 → D+1

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Submission text ?

Triangle begins:
   1:  5,  12,    4;
   2:  7,  30,   36,     8;
   3:  9,  56,  120,    96,    16;
   4: 11,  90,  280,   400,   240,     32;
   5: 13, 132,  540,  1120,  1200,    576,     64;
   6: 15, 182,  924,  2520,  3920,   3360,   1344,    128;
   7: 17, 240, 1456,  4928, 10080,  12544,   8960,   3072,    256;
   8: 19, 306, 2160,  8736, 22176,  36288,  37632,  23040,   6912,    512;
   9: 21, 380, 3060, 14400, 43680,  88704, 120960, 107520,  57600,  15360,  1024;
  10: 23, 462, 4180, 22440, 79200, 192192, 325248, 380160, 295680, 140800, 33792, 2048;

The row polynomials satisfy (2*n+1+2*x)*P(n,x) = (2*n+3+(4*n+8)*x+4*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 5 + 12*x + 4*x^2.

P[1, x_] := 5 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 3 + (4*n + 8)*x + 4*x^2)*P[n - 1, x])/(2*n + 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 5 + 12*x + 4*x^2; for(n = 2, N, my(i = n); v[i] = ((2*n + 3 + (4*n + 8)*x + 4*x^2)*v[i - 1])/(2*n + 1 + 2*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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 ?

• A156992 → basis A013611 → D+1
• A178822 → basis A013611 → D+1
• basis-map coefficients of A030528, shifted → basis A013609 → D+1
• A028326 → basis A013611 → D+1
• A120909 → D+1
• A178821 → basis A013611 → D+1
• A356546 → basis A013611 → D+1
• A130123 → basis A013609 → D+1
• A134082 → basis A013609
• A013609 → basis A013610 → D+1
• A178820 → basis A013611 → D+1
• A013609 → D+1
• A085880 → basis A013611 → D+1
• A136572 → basis A013609 → D+1
• basis-map coefficients of A026729 → basis A013609 → D+1
• A120909 → basis A013610 → D+1
• A038208 → basis A013611 → D+1
• A196347 → basis A013611 → D+1
• basis-map coefficients of A026729, shifted → basis A013609 → D+1
• A085881 → basis A013611 → D+1
• A038221 → basis A013611 → D+1
• A013609 → D → D+1

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.