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

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Triangle begins:
   1:  7,   24,     9;
   2: 10,   63,   108,     27;
   3: 13,  120,   378,    432,     81;
   4: 16,  195,   900,   1890,   1620,     243;
   5: 19,  288,  1755,   5400,   8505,    5832,     729;
   6: 22,  399,  3024,  12285,  28350,   35721,   20412,    2187;
   7: 25,  528,  4788,  24192,  73710,  136080,  142884,   69984,     6561;
   8: 28,  675,  7128,  43092, 163296,  398034,  612360,  551124,   236196,   19683;
   9: 31,  840, 10125,  71280, 323190,  979776, 1990170, 2624400,  2066715,  787320,   59049;
  10: 34, 1023, 13860, 111375, 588060, 2133054, 5388768, 9382230, 10825650, 7577955, 2598156, 177147;

The row polynomials satisfy (3*n+1+3*x)*P(n,x) = (3*n+4+(9*n+15)*x+9*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 7 + 24*x + 9*x^2.

P[1, x_] := 7 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 4 + (9*n + 15)*x + 9*x^2)*P[n - 1, x])/(3*n + 1 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 7 + 24*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = ((3*n + 4 + (9*n + 15)*x + 9*x^2)*v[i - 1])/(3*n + 1 + 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 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_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 ?

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