A099093 → x ↦ x−2

OEIS matches ?

• A000748: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:        1;
  1:       -6,        3;
  2:       30,      -33,         9;
  3:     -144,      252,      -144,       27;
  4:      684,    -1656,      1467,     -567,        81;
  5:    -3240,    10044,    -12150,     7209,     -2106,      243;
  6:    15336,   -57996,     89262,   -71901,     32076,    -7533,      729;
  7:   -72576,   324000,   -606528,   619488,   -373896,   133650,   -26244,    2187;
  8:   343440, -1767744,   3901608, -4837320,   3693681, -1782162,   531441,  -89667,    6561;
  9: -1625184,  9475056, -24101712, 35192232, -32572206, 19850427, -7976718, 2040471, -301806, 19683;

The row polynomials satisfy P(n,x) = -(6-3*x)*P(n-1,x) - (6-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -6 + 3*x.

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(6 - 3*x)*v[i - 1] - (6 - 3*x)*v[i - 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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (2/3, 2]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-8e25fd3c0a02e03b
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-8e25fd3c0a02e03b
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-8e25fd3c0a02e03b

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image affine, shift 0 e-8e25fd3c0a02e03b

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    • theorem_check ma_wang_interval e-a3711e6f8037cbc8 root: theorem check

      Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

• e-726dbe4e92e67f96 basis_map (A038207) → Riordan array (1, 3+3x).

Related sequences ?

No candidate equalities or related sequences.

References

1. [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