A099097 → x ↦ x−8

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:
   1:              1;
   2:            -24,             3;
   3:            568,          -143,              9;
   4:         -13440,          5088,           -642,           27;
   5:         318016,       -160720,          30457,        -2565,           81;
   6:       -7524864,       4757184,       -1202904,       152073,        -9612,        243;
   7:      178052608,    -135143232,       42736872,     -7207487,       683694,     -34587,        729;
   8:    -4213063680,    3732013056,    -1416734208,    298770816,    -37802148,    2869614,    -121014,     2187;
   9:    99689107456, -100948305920,    44720621952, -11320305440,   1790886961, -181316790,   11472759,  -414801,     6561;
  10: -2358834069504, 2687757496320, -1361073957888, 402042295680, -76341015384, 9663504783, -815458860, 44234991, -1399680, 19683;

The row polynomials satisfy P(n,x) = -(24-3*x)*P(n-1,x) - (8-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -24 + 3*x.

P[1, x_] := 1;
P[2, x_] := -24 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(24 - 3*x)*P[n - 1, x] - (8 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -24 + 3*x; for(n = 3, N, my(i = n); v[i] = -(24 - 3*x)*v[i - 1] - (8 - 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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (68/9, 8]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-7827a087c75bbe15
• Coefficients of one sign. Every nonzero row, n >= 1: 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-7827a087c75bbe15
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-7827a087c75bbe15

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-7827a087c75bbe15

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Riordan array (1, 3+x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
    • theorem_check ma_wang_interval e-09c939a271fe525a root: theorem check

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

Used in 1 proof step

• e-f43b3e2d606e8300 basis_map (A038279) → Riordan array (1, 3+x).

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