A274496 → D → x ↦ x+1

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Submission text ?

Triangle begins:
  0:    0;
  1:    0;
  2:    2;
  3:    4;
  4:   16,    8;
  5:   32,   16;
  6:   96,   96,  24;
  7:  192,  192,  48;
  8:  512,  768, 384,  64;
  9: 1024, 1536, 768, 128;

The row polynomials satisfy (n-2)*P(n,x) = (6*n-16+(n^2-3*n)*x)*P(n-1,x) - (16+(4*n+8)*x+2*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n - 16 + (n^2 - 3*n)*x)*P[n - 1, x] - (16 + (4*n + 8)*x + 2*n*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((6*n - 16 + (n^2 - 3*n)*x)*v[i - 1] - (16 + (4*n + 8)*x + 2*n*x^2)*deriv(v[i - 1], x))/(n - 2)); 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Laguerre’s theorem on differential operators
• 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

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from derivative of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal …?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …?
        • 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
• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from shift_x_plus_one of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry …?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …?
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …?
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A152198 → basis A013611
• A188440 → basis A013610
• A152198 → basis A013611 → basis A013610
• A274496 → basis A013611 → basis A013610
• A274496 → basis A013611 → D
• A274498 → D
• A103633 → x ↦ x/(x+1)
• A152198 → x ↦ x+1
• A274496 → basis A013611
• A152198 → basis A013609 → basis A013611
• A188440: Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
• A274498 → basis A013610
• A274498: Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k…
• A274496 → x ↦ x+1
• A152198 → basis A013609 → basis A007318
• A152198 → x ↦ x+1 → D
• A152198 → basis A013611 → D
• A274498 → D → D
• A188440 → D

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.