basis-map coefficients of A112227 → basis A013611 → D

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Triangle begins:
   1:  8,   32;
   2: 12,   96,   192;
   3: 16,  192,   768,   1024;
   4: 20,  320,  1920,   5120,    5120;
   5: 24,  480,  3840,  15360,   30720,    24576;
   6: 28,  672,  6720,  35840,  107520,   172032,   114688;
   7: 32,  896, 10752,  71680,  286720,   688128,   917504,   524288;
   8: 36, 1152, 16128, 129024,  645120,  2064384,  4128768,  4718592,   2359296;
   9: 40, 1440, 23040, 215040, 1290240,  5160960, 13762560, 23592960,  23592960,  10485760;
  10: 44, 1760, 31680, 337920, 2365440, 11354112, 37847040, 86507520, 129761280, 115343360, 46137344;

The row polynomials satisfy n*P(n,x) = (n+1+(4*n+4)*x)*P(n-1,x) for n >= 2, with P(1,x) = 8 + 32*x.

P[1, x_] := 8 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (4*n + 4)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 8 + 32*x; for(n = 2, N, my(i = n); v[i] = ((n + 1 + (4*n + 4)*x)*v[i - 1])/(n)); 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 ?

• basis-map coefficients of A026729, shifted → D → basis A013611
• A038231 → D → D
• image under θ+1 → basis A013611
• A013611 → D → D
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• A131431 → basis A013611
• A136572 → basis A013611 → reverse
• A130123 → basis A013611 → D
• A027466 → basis A013609 → D
• basis-map coefficients of A026729, shifted → θ+1 → basis A013611
• A136572 → (θ+1)² → basis A013611
• A136572 → D → basis A013611
• image under θ+1 → θ+1 → D → basis A013611
• A130123 → (θ+1)² → basis A013611
• basis-map coefficients of A021012 → basis A013611
• A038329 → reverse
• A027467 → basis A013611 → reverse
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A027466 → basis A013609
• A136521 → basis A013611
• A136572 → θ+1 → basis A013611
• basis-map coefficients of A112227 → basis A013611
• A131431 → θ+1 → basis A013611
• A027467 → basis A013611 → D
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• image under θ+1 → (θ+1)² → basis A013611
• A038330 → basis A007318
• A038329 → D
• A136572 → basis A013611
• basis-map coefficients of A030528, shifted → D → basis A013611
• A130123 → D → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A131436 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → reverse
• A130123 → basis A013611
• A127648 → D → basis A013611
• A130123 → θ+1 → basis A013611
• A038303 → basis A007318 → basis A013610
• A013610 → basis A013611 → x ↦ x/(x+1)
• A120910 → basis A007318 → basis A007318
• A132440 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → D
• basis-map coefficients of A112227 → θ+1 → basis A013611
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• A134309 → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A038231 → D
• A129186 → basis A013611
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• basis-map coefficients of A021012 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• A131431 → D → basis A013611
• A132440 → θ+1 → basis A013611
• basis-map coefficients of A026729 → D → basis A013611
• A038330 → basis A007318 → D
• A013611 → D
• basis-map coefficients of A026729 → basis A013611 → reverse
• A027467 → basis A013611
• A038330 → basis A007318 → reverse
• A136572 → basis A013611 → D
• basis-map coefficients of A030528, shifted → basis A013611

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.