A274496 → basis A013611 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:     1;
  1:     2;
  2:    12,      8;
  3:    24,     16;
  4:   144,    192,     64;
  5:   288,    384,    128;
  6:  1728,   3456,   2304,   512;
  7:  3456,   6912,   4608,  1024;
  8: 20736,  55296,  55296, 24576, 4096;
  9: 41472, 110592, 110592, 49152, 8192;

The row polynomials satisfy P(n,x) = (4*n-2+(4*n-4)*x)*P(n-1,x) - (12+20*x+8*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n - 2 + (4*n - 4)*x)*P[n - 1, x] - (12 + 20*x + 8*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n - 2 + (4*n - 4)*x)*v[i - 1] - (12 + 20*x + 8*x^2)*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 (−∞, -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), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry…?
    • 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 [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
• Ore-algebra identity of two definitions polytool ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry…?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A274496 → basis A013610
• A188440 → basis A007318 → D
• A152198 → basis A013610
• A152198 → basis A013610 → D
• A152198 → basis A013611 → basis A007318
• A274498 → x ↦ x+1
• A274496 → basis A013610 → D
• A188440 → basis A007318
• A152198 → basis A013609 → basis A013610
• A274498 → x ↦ x+1 → 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.