A274496 → D → D

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Triangle begins:
  0:   0;
  1:   0;
  2:   0;
  3:   0;
  4:   8;
  5:  16;
  6:  48,  48;
  7:  96,  96;
  8: 192, 384, 192;
  9: 384, 768, 384;

The row polynomials satisfy (n-4)*P(n,x) = -(n^2-15*n+48-(n^2-5*n)*x)*P(n-1,x) + (2*n-16-16*x-2*n*x^2)*d/dx P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 0, P(4,x) = 8.

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 0;
P[4, x_] := 8;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 15*n + 48 - (n^2 - 5*n)*x)*P[n - 1, x] + (2*n - 16 - 16*x - 2*n*x^2)*D[P[n - 1, x], x])/(n - 4)]];
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] = 0; v[4] = 0; v[5] = 8; for(n = 5, N, my(i = n + 1); v[i] = (-(n^2 - 15*n + 48 - (n^2 - 5*n)*x)*v[i - 1] + (2*n - 16 - 16*x - 2*n*x^2)*deriv(v[i - 1], x))/(n - 4)); 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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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

Related ?

Same fingerprint ?

• A103633: Triangle read by rows: triangle of repeated stepped binomial coefficients.
• A152198 → basis A013609 → basis A013609
• A274496 → basis A013609
• A152198 → D
• A274496 → D
• A152198 → basis A013609
• A152198 → basis A013609 → D
• A132774 → Veronese 2 → x ↦ x+1
• A274496: Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k …
• A152198: Triangle read by rows, A007318 rows repeated.
• A132774 → Veronese 2 (j=1) → x ↦ x+1

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.