basis-map coefficients of A112227 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
   1: 1, 12,   27;
   2: 1, 18,   81,   108;
   3: 1, 24,  162,   432,    405;
   4: 1, 30,  270,  1080,   2025,   1458;
   5: 1, 36,  405,  2160,   6075,   8748,    5103;
   6: 1, 42,  567,  3780,  14175,  30618,   35721,   17496;
   7: 1, 48,  756,  6048,  28350,  81648,  142884,  139968,   59049;
   8: 1, 54,  972,  9072,  51030, 183708,  428652,  629856,  531441,   196830;
   9: 1, 60, 1215, 12960,  85050, 367416, 1071630, 2099520, 2657205,  1968300,  649539;
  10: 1, 66, 1485, 17820, 133650, 673596, 2357586, 5773680, 9743085, 10825650, 7144929, 2125764;

The row polynomials satisfy (1+(3*n+3)*x)*P(n,x) = (1+(3*n+9)*x+(9*n+18)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 12*x + 27*x^2.

P[1, x_] := 1 + 12*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n + 9)*x + (9*n + 18)*x^2)*P[n - 1, x])/(1 + (3*n + 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 12*x + 27*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + (3*n + 9)*x + (9*n + 18)*x^2)*v[i - 1])/(1 + (3*n + 3)*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_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)

Related ?

Same fingerprint ?

• basis-map coefficients of A030528, shifted → basis A013610 → θ+1
• A013610 → D → θ+1
• A013610 → θ+1
• A013610 → basis A013611 → θ+1
• A130123 → basis A013610 → θ+1
• basis-map coefficients of A026729 → basis A013610 → θ+1
• basis-map coefficients of A026729, shifted → basis A013610 → θ+1
• A136572 → basis A013610 → 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.