Real-rooted atlas

A060920: Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).

A060920 nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A060920
\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
255111
313209143
4347151141171
589235233105201683
62337449425941902712731
7610228535222860129531535110923

Show all 15 stored rows

Generated b-files: A007583 (row sums), A060920 ?

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-6 - 5\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(3 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
Technical details
Node id
n-fac4d4a000d744e8 · revision 2 · JSON
Definition source
oeis_catalog:A060920
derived:shift_x_plus_one (equal by e-bc0a3892b1359edd)
derived:basis_map_A007318 (equal by e-d8ffe0ceb9f858f8)
Fingerprint
f6-5870c1c52d074a69

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:    2,     1;
  2:    5,     5,     1;
  3:   13,    20,     9,     1;
  4:   34,    71,    51,    14,     1;
  5:   89,   235,   233,   105,    20,     1;
  6:  233,   744,   942,   594,   190,    27,    1;
  7:  610,  2285,  3522,  2860,  1295,   315,   35,   1;
  8: 1597,  6865, 12473, 12402,  7285,  2534,  490,  44,  1;
  9: 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] - (1 + 2*x + x^2)*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] - (1 + 2*x + x^2)*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.

A060920: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    1;
  1:    2,     1;
  2:    5,     5,     1;
  3:   13,    20,     9,     1;
  4:   34,    71,    51,    14,     1;
  5:   89,   235,   233,   105,    20,     1;
  6:  233,   744,   942,   594,   190,    27,    1;
  7:  610,  2285,  3522,  2860,  1295,   315,   35,   1;
  8: 1597,  6865, 12473, 12402,  7285,  2534,  490,  44,  1;
  9: 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) - (1+2*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] - (1 + 2*x + x^2)*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] - (1 + 2*x + x^2)*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-1b409596bf82290e
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f409e7a544612dfe
  • 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-5dc8ce8b8b471ab9, e-f409e7a544612dfe
  • 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-5dc8ce8b8b471ab9, e-f409e7a544612dfe
  • 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-5dc8ce8b8b471ab9, e-f409e7a544612dfe
  • 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-5dc8ce8b8b471ab9

Proof chains

5 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 9 proof steps

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

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