Real-rooted atlas

A358735: 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*x + 2)*a(m+1) - a(m) for all m in Z.

A358735 nonnegative coefficients

Proved for all n: all roots real, in (−2, 0) · interlacing · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A358735
\(P_{n}(x) = \left(2 + n x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21427
311016633
4120707624191
51352244964281201304
615658822603808280872010241
718413448140230083215221096504090865

Show all 15 stored rows

Generated b-files: A358735 ?

Computed in your browser ?

Technical details
Node id
n-f3e0631eb8740f76 · revision 1 · JSON
Definition source
oeis_catalog:A358735
Fingerprint
f6-b3267034d12950ac

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

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.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,    2;
  3: 1,  10,   16,     6;
  4: 1,  20,   70,    76,     24;
  5: 1,  35,  224,   496,    428,     120;
  6: 1,  56,  588,  2260,   3808,    2808,     720;
  7: 1,  84, 1344,  8140,  23008,   32152,   21096,    5040;
  8: 1, 120, 2772, 24772, 107328,  245560,  298688,  178848,   40320;
  9: 1, 165, 5280, 66352, 414596, 1424920, 2786320, 3040848, 1690272, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (2+n*x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + n*x)*P[n - 1, x] - P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + n*x)*v[i - 1] - v[i - 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)

Generated with polytool c48629374597.

A358735: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,    2;
  3: 1,  10,   16,     6;
  4: 1,  20,   70,    76,     24;
  5: 1,  35,  224,   496,    428,     120;
  6: 1,  56,  588,  2260,   3808,    2808,     720;
  7: 1,  84, 1344,  8140,  23008,   32152,   21096,    5040;
  8: 1, 120, 2772, 24772, 107328,  245560,  298688,  178848,   40320;
  9: 1, 165, 5280, 66352, 414596, 1424920, 2786320, 3040848, 1690272, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (2+n*x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + n*x)*P[n - 1, x] - P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + n*x)*v[i - 1] - v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-db65c5b95cec1d15
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-67c88de355bdef8b, e-a98b0ab5851394b4, e-db65c5b95cec1d15
  • 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-a98b0ab5851394b4
  • 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-67c88de355bdef8b, e-a98b0ab5851394b4
  • 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-67c88de355bdef8b, e-a98b0ab5851394b4
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-67c88de355bdef8b

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

Used in 7 proof steps

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