A099093 → x ↦ x−7

OEIS matches ?

No OEIS matches.

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:           -21,            3;
  2:           420,         -123,             9;
  3:         -8379,         3717,          -549,           27;
  4:        167139,       -99351,         22122,        -2187,           81;
  5:      -3333960,      2484594,       -739935,       110079,        -8181,        243;
  6:      66503241,    -59590566,      22229802,     -4419171,       493776,     -29403,        729;
  7:   -1326554901,   1388733255,    -622605123,    154960533,    -23124771,    2069145,    -102789,     2187;
  8:   26461084860, -31692151449,   16595309808,  -4962494565,    926874981, -110727567,    8262486,  -352107,     6561;
  9: -527825129139, 711775371951, -426337052967, 148876328727, -33401356506, 4993077492, -497328903, 31827411, -1187541, 19683;

The row polynomials satisfy P(n,x) = -(21-3*x)*P(n-1,x) - (21-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -21 + 3*x.

P[0, x_] := 1;
P[1, x_] := -21 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(21 - 3*x)*P[n - 1, x] - (21 - 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] = -21 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(21 - 3*x)*v[i - 1] - (21 - 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 (17/3, 7]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-bc143c20b95292bc
• 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-bc143c20b95292bc
• 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-bc143c20b95292bc

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-bc143c20b95292bc

  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-04516916870a85c6 basis_map (A027466) → 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