basis-map coefficients of A112227 → x ↦ x+1 → basis A008277

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Triangle begins:
   1:    4,      5,       1;
   2:    8,     19,       9,       1;
   3:   16,     65,      55,      14,       1;
   4:   32,    211,     285,     125,      20,       1;
   5:   64,    665,    1351,     910,     245,      27,       1;
   6:  128,   2059,    6069,    5901,    2380,     434,      35,      1;
   7:  256,   6305,   26335,   35574,   20181,    5418,     714,     44,     1;
   8:  512,  19171,  111645,  204205,  156660,   58107,   11130,   1110,    54,    1;
   9: 1024,  58025,  465751, 1132670, 1144165,  563409,  147147,  21120,  1650,   65,  1;
  10: 2048, 175099, 1921029, 6129101, 7997660, 5088028, 1740585, 337227, 37620, 2365, 77, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 5*x + x^2.

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

P(N) = {my(v = vector(N)); v[1] = 4 + 5*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + x*deriv(v[i - 1], 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 Ray basis map theorem, 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 Ray basis map theorem, Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem, 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

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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 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 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)
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A008277 of basis-map coefficients of A scaled Hermite triangle.?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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)

Related ?

Same fingerprint ?

• A134058 → basis A008277
• A028326 → basis A013609 → basis A008277
• A178821 → basis A008277
• A038338 → basis A008277
• A085880 → basis A008277
• A038325 → basis A008277
• A126351 → reverse
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• basis-map coefficients of A021012 → basis A008277 → D+1
• A027465 → basis A013611 → basis A008277
• A132200 → basis A008277
• A003506 → basis A008277
• A196347 → basis A008277
• A178822 → basis A013609 → basis A008277
• A120910 → basis A013610 → basis A008277
• A085880 → basis A013609 → basis A008277
• A135838 → basis A008277
• A124860 → basis A008277
• A356546 → basis A013609 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A143494 → reverse
• A131431 → x ↦ x+1 → basis A008277
• A156992 → basis A008277 → reverse
• A135089 → basis A008277
• basis-map coefficients of A026729, shifted → basis A007318 → basis A008277
• A178821 → basis A013609 → basis A008277
• A156992 → basis A008277
• basis-map coefficients of A113214, shifted → D → basis A008277
• A376827 → basis A008277
• A085880 → D → basis A008277
• A028326 → basis A008277
• A258758 → basis A008277
• A038338 → basis A013609 → basis A008277
• A178820 → basis A013609 → basis A008277
• A156992 → basis A013609 → basis A008277
• A135065 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A038325 → basis A013609 → basis A008277
• A028326 → D → basis A008277
• A038221 → basis A008277
• A134059 → basis A008277
• A178820 → basis A008277
• basis-map coefficients of A113214, shifted → basis A013609 → basis A008277
• A085881 → basis A008277
• A038221 → basis A013609 → basis A008277
• A196347 → basis A013609 → basis A008277
• A038208 → D → basis A008277
• A178822 → basis A008277
• A038330 → basis A013611 → basis A008277
• A085881 → basis A013609 → basis A008277
• A344565 → basis A008277
• A094305 → basis A008277
• A134346 → basis A008277
• basis-map coefficients of A127647 → basis A008277
• A356546 → basis A008277
• A038208 → basis A008277
• A003506 → D → basis A008277

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.