A327997 → basis A013610

OEIS matches ?

No OEIS matches.

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,        3;
  2:      17,       27,         9;
  3:      78,      195,       135,        27;
  4:     393,     1356,      1449,       594,        81;
  5:    2208,     9687,     14040,      8775,      2430,      243;
  6:   13977,    73467,    133686,    112185,     46980,     9477,      729;
  7:  100026,   602235,   1304541,   1364958,    762615,   231336,    35721,    2187;
  8:  806769,  5381856,  13349250,  16572276,  11497059,  4633524,  1071630,  131220,   6561;
  9: 7280604, 52548939, 145027584, 206156070, 169378776, 84541401, 25902828, 4737042, 472392, 19683;

The row polynomials satisfy P(n,x) = -(3*n-3)*P(n-2,x) + (n+3+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(3*n - 3)*P[n - 2, x] + (n + 3 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*n - 3)*v[i - 2] + (n + 3 + 3*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-5617fd17362674c9
• 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-5617fd17362674c9
• 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-5617fd17362674c9
• 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-5617fd17362674c9

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-5617fd17362674c9

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-a526bc158f95a9e3 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