A091042 → basis A013610

OEIS matches ?

• A096053: 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:
   1:      1;
   2:      4,       9;
   3:     16,      60,       45;
   4:     64,     336,      504,      189;
   5:    256,    1728,     3888,     3240,       729;
   6:   1024,    8448,    25344,    33264,     17820,      2673;
   7:   4096,   39936,   149760,   269568,    235872,     88452,      9477;
   8:  16384,  184320,   829440,  1900800,   2332800,   1469664,    408240,    32805;
   9:  65536,  835584,  4386816, 12220416,  19388160,  17449344,   8328096,  1784592,  111537;
  10: 262144, 3735552, 22413312, 73543680, 143410176, 169019136, 117013248, 43879968, 7479540, 373977;

The row polynomials satisfy P(n,x) = -9*x^2*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 9*x.

P[1, x_] := 1;
P[2, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[-9*x^2*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 9*x; for(n = 3, N, my(i = n); v[i] = -9*x^2*v[i - 2] + (4 + 6*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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-f9693f57c5cb4021
• Coefficients of one sign. Every nonzero row, n >= 1: 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-f9693f57c5cb4021
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-f9693f57c5cb4021
• 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-f9693f57c5cb4021

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

• basis_map A013610 e-f9693f57c5cb4021

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-1c0df90858517049 root: theorem check

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

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