A099097 → x ↦ x−15

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:              -45,               3;
   3:             2010,            -269,                9;
   4:           -89775,           18045,            -1209,             27;
   5:          4009725,        -1075305,           108136,          -4833,             81;
   6:       -179091000,        60057450,         -8055855,         540279,         -18117,          243;
   7:       7998949125,     -3219718950,        539988480,      -48299489,        2430054,       -65205,         729;
   8:    -357266345625,    167804247375,     -33777743175,     3777282405,     -253438863,     10202625,     -228177,      2187;
   9:   15957001316250,  -8566695435375,    2012091638850,  -270046456935,    22651845751,  -1216026585,    40799700,   -782217,     6561;
  10: -712706064046875, 430497968484375, -115569739659375, 18097928499375, -1821893564250, 122270255340, -5470440975, 157337883, -2639709, 19683;

The row polynomials satisfy P(n,x) = -(45-3*x)*P(n-1,x) - (15-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -45 + 3*x.

P[1, x_] := 1;
P[2, x_] := -45 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(45 - 3*x)*P[n - 1, x] - (15 - 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] = -45 + 3*x; for(n = 3, N, my(i = n); v[i] = -(45 - 3*x)*v[i - 1] - (15 - 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 (131/9, 15]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-6b5eff9154c3c40c
• 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-6b5eff9154c3c40c
• 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-6b5eff9154c3c40c

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-6b5eff9154c3c40c

  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-ae6dc95f51d15092 basis_map (A027467) → 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