A110441 → basis A007318

OEIS matches ?

• A107839: row sums, 15 terms agree OEIS nonn (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:
  0:     1;
  1:     4,      1;
  2:    14,      8,      1;
  3:    48,     44,     12,      1;
  4:   164,    208,     90,     16,     1;
  5:   560,    908,    544,    152,    20,     1;
  6:  1912,   3776,   2904,   1120,   230,    24,    1;
  7:  6528,  15200,  14304,   7080,  2000,   324,   28,   1;
  8: 22288,  59776,  66608,  40384, 14620,  3248,  434,  32,  1;
  9: 76096, 230992, 297600, 213984, 94864, 26964, 4928, 560, 36, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (4 + x)*v[i - 1]); 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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-1d7f8f614531604f
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n 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-1d7f8f614531604f, e-5bec3e19f7ad29fa
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-1d7f8f614531604f, e-5bec3e19f7ad29fa
• 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-1d7f8f614531604f, e-5bec3e19f7ad29fa
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-7, -1)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5bec3e19f7ad29fa

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-1d7f8f614531604f

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangular array formed by the Mersenne numbers.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-c01294f1353c0463 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

    • theorem_check three_term_recurrence e-02d1e2cf79c791bd root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

• definition_equality e-103d916d7cdca8d3 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-5bec3e19f7ad29fa

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array formed by the Mersenne numbers.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, 0)
    • ↵ theorem_check three_term_recurrence e-02d1e2cf79c791bd (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023