A207606 → basis A013609

OEIS matches ?

• A002310: 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:    13,     22,       8;
  1:    34,     88,      68,      16;
  2:    89,    310,     372,     184,       32;
  3:   233,   1020,    1668,    1280,      464,       64;
  4:   610,   3216,    6672,    6992,     3920,     1120,      128;
  5:  1597,   9848,   24780,   33040,    25280,    11136,     2624,     256;
  6:  4181,  29522,   87364,  141688,   138000,    82848,    30016,    6016,     512;
  7: 10946,  87080,  296356,  566752,   672096,   513408,   253120,   77824,   13568,   1024;
  8: 28657, 253610,  975864, 2151280,  3011792,  2801568,  1756160,  733696,  195840,  30208,  2048;
  9: 75025, 731064, 3138456, 7838816, 12665840, 13914880, 10618496, 5635584, 2041344, 481280, 66560, 4096;

The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 22*x + 8*x^2, P(1,x) = 34 + 88*x + 68*x^2 + 16*x^3.

P[0, x_] := 13 + 22*x + 8*x^2;
P[1, x_] := 34 + 88*x + 68*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 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] = 13 + 22*x + 8*x^2; v[2] = 34 + 88*x + 68*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + 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-0adfae28f720d284
• 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-0adfae28f720d284
• 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-0adfae28f720d284
• 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-0adfae28f720d284

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 A013609 e-0adfae28f720d284

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-62ac8b7225a515c7 root: theorem check

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

    • theorem_check three_term_recurrence e-3222de37fcca30b3 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