A358735 → basis A008288

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:       2,         1;
  2:       7,        10,         2;
  3:      33,        88,        46,          6;
  4:     191,       778,       762,        244,         24;
  5:    1304,      7263,     11268,       6492,       1508,       120;
  6:   10241,     72968,    161112,     144476,      58600,     10728,       720;
  7:   90865,    792816,   2317324,    2952204,    1827472,    571648,     86616,     5040;
  8:  898409,   9309024,  34244032,   58117724,   50577688,  23754776,   6050912,   783648,   40320;
  9: 9791634, 117777801, 525759196, 1133657668, 1309445980, 855140280, 322796224, 69400368, 7859232, 362880;

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (n^3+n^2-4*n+(n^3-3*n^2+2*n)*x)*P(n-3,x) - (2*n^3-8*n+2-(n^4-6*n^3+11*n^2-6*n)*x)*P(n-2,x) + (n^3+n^2-8*n+4+(n^3-3*n^2+2*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 7 + 10*x + 2*x^2, P(3,x) = 33 + 88*x + 46*x^2 + 6*x^3.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 7 + 10*x + 2*x^2;
P[3, x_] := 33 + 88*x + 46*x^2 + 6*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (n^3 + n^2 - 4*n + (n^3 - 3*n^2 + 2*n)*x)*P[n - 3, x] - (2*n^3 - 8*n + 2 - (n^4 - 6*n^3 + 11*n^2 - 6*n)*x)*P[n - 2, x] + (n^3 + n^2 - 8*n + 4 + (n^3 - 3*n^2 + 2*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 7 + 10*x + 2*x^2; v[4] = 33 + 88*x + 46*x^2 + 6*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (n^3 + n^2 - 4*n + (n^3 - 3*n^2 + 2*n)*x)*v[i - 3] - (2*n^3 - 8*n + 2 - (n^4 - 6*n^3 + 11*n^2 - 6*n)*x)*v[i - 2] + (n^3 + n^2 - 8*n + 4 + (n^3 - 3*n^2 + 2*n)*x)*v[i - 1])/(n^2 - 3*n + 2)); 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 [-1564556493/268435456, -46056243/268435456]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-393d995dcff5ebfb
• 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-393d995dcff5ebfb, e-fca58ee161653196
• 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-393d995dcff5ebfb, e-fca58ee161653196
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-fca58ee161653196

Proof chains

2 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 A008288 e-393d995dcff5ebfb

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [-2, 0]
    • theorem_check three_term_recurrence e-67c88de355bdef8b root: theorem check

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

• basis_map A008288 e-fca58ee161653196

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [-2, 0]
    • ↵ theorem_check three_term_recurrence e-67c88de355bdef8b (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