basis-map coefficients of A112227 → basis A013611 → θ+1

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Submission text ?

Triangle begins:
   1: 1, 16,   48;
   2: 1, 24,  144,   256;
   3: 1, 32,  288,  1024,   1280;
   4: 1, 40,  480,  2560,   6400,    6144;
   5: 1, 48,  720,  5120,  19200,   36864,    28672;
   6: 1, 56, 1008,  8960,  44800,  129024,   200704,   131072;
   7: 1, 64, 1344, 14336,  89600,  344064,   802816,  1048576,   589824;
   8: 1, 72, 1728, 21504, 161280,  774144,  2408448,  4718592,  5308416,   2621440;
   9: 1, 80, 2160, 30720, 268800, 1548288,  6021120, 15728640, 26542080,  26214400,  11534336;
  10: 1, 88, 2640, 42240, 422400, 2838528, 13246464, 43253760, 97320960, 144179200, 126877696, 50331648;

The row polynomials satisfy (1+(4*n+4)*x)*P(n,x) = (1+(4*n+12)*x+(16*n+32)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 16*x + 48*x^2.

P[1, x_] := 1 + 16*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (4*n + 12)*x + (16*n + 32)*x^2)*P[n - 1, x])/(1 + (4*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 16*x + 48*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + (4*n + 12)*x + (16*n + 32)*x^2)*v[i - 1])/(1 + (4*n + 4)*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013611 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_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 (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (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)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → basis A013611 → θ+1
• A013611 → θ+1
• A013611 → D → θ+1
• A130123 → basis A013611 → θ+1
• A136572 → basis A013611 → θ+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.