basis-map coefficients of A112227 → basis A008277 → D

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Submission text ?

Triangle begins:
   1:    3,      2;
   2:    7,     12,       3;
   3:   15,     50,      30,       4;
   4:   31,    180,     195,      60,       5;
   5:   63,    602,    1050,     560,     105,       6;
   6:  127,   1932,    5103,    4200,    1330,     168,       7;
   7:  255,   6050,   23310,   27804,   13230,    2772,     252,      8;
   8:  511,  18660,  102315,  170100,  114135,   35280,    5250,    360,     9;
   9: 1023,  57002,  437250,  986920,  897435,  383922,   83160,   9240,   495,  10;
  10: 2047, 173052, 1834503, 5517600, 6618260, 3764376, 1113189, 178200, 15345, 660, 11;

The row polynomials satisfy P(n,x) = -(2+2*x+x^2)*P(n-2,x) - (x+x^2)*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 3 + 2*x, P(2,x) = 7 + 12*x + 3*x^2.

P[1, x_] := 3 + 2*x;
P[2, x_] := 7 + 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(2 + 2*x + x^2)*P[n - 2, x] - (x + x^2)*D[P[n - 2, x], x] + (3 + 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] = 3 + 2*x; v[2] = 7 + 12*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = -(2 + 2*x + x^2)*v[i - 2] - (x + x^2)*deriv(v[i - 2], x) + (3 + 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 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_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 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)

Related ?

Same fingerprint ?

• A008277 → D
• basis-map coefficients of A021012 → basis A008277 → D
• A008277 → θ+1 → D+1
• A130123 → basis A008277 → D
• A008277 → θ+1 → D+1
• A008277 → θ+1 → D+1 → reverse

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.