A063967 → x ↦ x−5

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:
   1:     1;
   2:    -4,     1;
   3:    12,    -7,      1;
   4:   -32,    32,    -10,      1;
   5:    80,  -120,     61,    -13,      1;
   6:  -192,   400,   -292,     99,    -16,     1;
   7:   448, -1232,   1204,   -575,    146,   -19,     1;
   8: -1024,  3584,  -4480,   2816,   -996,   202,   -22,   1;
   9:  2304, -9984,  15456, -12240,   5641, -1582,   267, -25,   1;
  10: -5120, 26880, -50304,  48672, -28004, 10165, -2360, 341, -28, 1;

The row polynomials satisfy P(n,x) = -(4-x)*P(n-1,x) - (4-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -4 + x.

P[1, x_] := 1;
P[2, x_] := -4 + 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, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -4 + x; for(n = 3, N, my(i = n); 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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-81aeef92852c73bb
• 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-81aeef92852c73bb
• 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-81aeef92852c73bb

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-81aeef92852c73bb

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
    • theorem_check ma_wang_interval e-32a5b77539bbfdf5 root: theorem check

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

Used in 1 proof step

• e-a054677672f9106d basis_map (A038243) → Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) …

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