A152830 → x ↦ x+1 → basis A013610

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Submission text ?

Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1;
  4: 3,  3;
  5: 3,  3;
  6: 3,  3;
  7: 3,  3;
  8: 9, 18, 9;
  9: 9, 18, 9;

The row polynomials satisfy P(n,x) = (3+3*x)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1, P(3,x) = 1.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from shift_x_plus_one of Triangle read by rows, A007318 rows repeated four times .?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, A007318 rows repeated four times .?
        • 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
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows, A007318 rows repeated four times .?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A152830: Triangle read by rows, A007318 rows repeated four times .
• A152830 → basis A013609
• A152830 → basis A013609 → basis A013609

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.