A091044 → basis A007318

OEIS matches ?

• A001109: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,       2;
  2:     16,      16,       3;
  3:     64,      96,      40,       4;
  4:    256,     512,     336,      80,       5;
  5:   1024,    2560,    2304,     896,     140,      6;
  6:   4096,   12288,   14080,    7680,    2016,    224,      7;
  7:  16384,   57344,   79872,   56320,   21120,   4032,    336,     8;
  8:  65536,  262144,  430080,  372736,  183040,  50688,   7392,   480,   9;
  9: 262144, 1179648, 2228224, 2293760, 1397760, 512512, 109824, 12672, 660, 10;

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

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] + (4 + 2*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] + (4 + 2*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-24d6e61d1fff2e02
• 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-24d6e61d1fff2e02, e-c620f77b38a31b20
• 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-24d6e61d1fff2e02, e-c620f77b38a31b20
• 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-24d6e61d1fff2e02, e-c620f77b38a31b20
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-c620f77b38a31b20

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-24d6e61d1fff2e02

  Geometric basis map (elementary) polytool ?

  • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4ffbabf9b60188a8 root: theorem check

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

    • preserver_image reversal, shift 0 e-75f779345b39f910

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)
• definition_equality e-507237b5f4039072 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-c620f77b38a31b20

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check three_term_recurrence e-4ffbabf9b60188a8 (expanded above)
    • ↵ preserver_image reversal, shift 0 e-75f779345b39f910 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A091044 → x ↦ x/(x+1)

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