A113214 → x ↦ x−4

OEIS matches ?

• A123877: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A123876: triangle, 55 terms agree, 10 full rows row-aligned (tabl) · view at the OEIS · generated b-file 990 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:   -2,     1;
  2:    4,    -5,      1;
  3:   -8,    18,     -8,     1;
  4:   16,   -56,     41,   -11,      1;
  5:  -32,   160,   -170,    73,    -14,     1;
  6:   64,  -432,    620,  -377,    114,   -17,     1;
  7: -128,  1120,  -2072,  1666,   -704,   164,   -20,   1;
  8:  256, -2816,   6496, -6608,   3649, -1178,   223, -23,   1;
  9: -512,  6912, -19392, 24192, -16722,  7001, -1826, 291, -26, 1;

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

P[0, x_] := 1;
P[1, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[-(4 - x)*P[n - 1, x] - (4 - 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] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(4 - x)*v[i - 1] - (4 - 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)

Triangle begins:
  0:    1;
  1:   -2,     1;
  2:    4,    -5,      1;
  3:   -8,    18,     -8,     1;
  4:   16,   -56,     41,   -11,      1;
  5:  -32,   160,   -170,    73,    -14,     1;
  6:   64,  -432,    620,  -377,    114,   -17,     1;
  7: -128,  1120,  -2072,  1666,   -704,   164,   -20,   1;
  8:  256, -2816,   6496, -6608,   3649, -1178,   223, -23,   1;
  9: -512,  6912, -19392, 24192, -16722,  7001, -1826, 291, -26, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(4 - x)*v[i - 1] - (4 - x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 (0, 4]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-85e6814e89c84b8b
• 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-85e6814e89c84b8b
• 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-85e6814e89c84b8b

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-85e6814e89c84b8b

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    • theorem_check ma_wang_interval e-5a8f629e61d609b9 root: theorem check

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

Used in 1 proof step

• e-a363e9242012bc21 basis_map (A038231) → Riordan array (1+2x,x(1+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