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

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Submission text ?

Triangle begins:
   1:  9,   40,    16;
   2: 13,  108,   240,     64;
   3: 17,  208,   864,   1280,     256;
   4: 21,  340,  2080,   5760,    6400,     1024;
   5: 25,  504,  4080,  16640,   34560,    30720,     4096;
   6: 29,  700,  7056,  38080,  116480,   193536,   143360,    16384;
   7: 33,  928, 11200,  75264,  304640,   745472,  1032192,   655360,     65536;
   8: 37, 1188, 16704, 134400,  677376,  2193408,  4472832,  5308416,   2949120,    262144;
   9: 41, 1480, 23760, 222720, 1344000,  5419008, 14622720, 25559040,  26542080,  13107200,  1048576;
  10: 45, 1804, 32560, 348480, 2449920, 11827200, 39739392, 91914240, 140574720, 129761280, 57671680, 4194304;

The row polynomials satisfy (4*n+1+4*x)*P(n,x) = (4*n+5+(16*n+24)*x+16*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 9 + 40*x + 16*x^2.

P[1, x_] := 9 + 40*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 5 + (16*n + 24)*x + 16*x^2)*P[n - 1, x])/(4*n + 1 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 9 + 40*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = ((4*n + 5 + (16*n + 24)*x + 16*x^2)*v[i - 1])/(4*n + 1 + 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 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_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 ?

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