A130123 → basis A008277 → D

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Submission text ?

Triangle begins:
  0:      0;
  1:      2;
  2:     12,       8;
  3:     56,      96,       24;
  4:    240,     800,      480,       64;
  5:    992,    5760,     6240,     1920,      160;
  6:   4032,   38528,    67200,    35840,     6720,      384;
  7:  16256,  247296,   653184,   537600,   170240,    21504,     896;
  8:  65280, 1548800,  5967360,  7117824,  3386880,   709632,   64512,   2048;
  9: 261632, 9553920, 52385280, 87091200, 58437120, 18063360, 2688000, 184320, 4608;

The row polynomials satisfy P(n,x) = -(8+8*x+4*x^2)*P(n-2,x) - (4*x+4*x^2)*d/dx P(n-2,x) + (6+4*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(8 + 8*x + 4*x^2)*P[n - 2, x] - (4*x + 4*x^2)*D[P[n - 2, x], x] + (6 + 4*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 8*x + 4*x^2)*v[i - 2] - (4*x + 4*x^2)*deriv(v[i - 2], x) + (6 + 4*x)*v[i - 1] + 2*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 - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A008277 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,…?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

      • from Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A021012 → basis A008277 → D
• A008277 → θ+1 → D+1
• A008277 → θ+1 → D+1
• basis-map coefficients of A112227 → basis A008277 → D
• A008277 → D
• 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.