Real-rooted atlas

A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.

A111125 nonnegative coefficients

Proved for all n: all roots real, in (−4, 0) · interlacing · coefficients of one sign · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A111125
\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
355111
47147129
5930279176
611557744111199
7139118215665131521
815140378450275901511364

Show all 15 stored rows

Generated b-files: A002878 (row sums), A082985 (reversed triangle), A111125 ?

Computed in your browser ?

Technical details
Node id
n-318700a428d7b97b · revision 1 · JSON
Definition source
oeis_catalog:A111125
Fingerprint
f6-fb7ccb346707daea

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  3,   1;
   3:  5,   5,    1;
   4:  7,  14,    7,    1;
   5:  9,  30,   27,    9,    1;
   6: 11,  55,   77,   44,   11,    1;
   7: 13,  91,  182,  156,   65,   13,   1;
   8: 15, 140,  378,  450,  275,   90,  15,   1;
   9: 17, 204,  714, 1122,  935,  442, 119,  17,  1;
  10: 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) - P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] - P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = (2 + 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, v[i])))};
T(10)

Generated with polytool c48629374597.

A082985: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  5,   5;
  3: 1,  7,  14,   7;
  4: 1,  9,  27,  30,    9;
  5: 1, 11,  44,  77,   55,   11;
  6: 1, 13,  65, 156,  182,   91,   13;
  7: 1, 15,  90, 275,  450,  378,  140,   15;
  8: 1, 17, 119, 442,  935, 1122,  714,  204,  17;
  9: 1, 19, 152, 665, 1729, 2717, 2508, 1254, 285, 19;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] - P[n - 2, x]];
Table[Reverse[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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] - v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A111125: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  3,   1;
  2:  5,   5,    1;
  3:  7,  14,    7,    1;
  4:  9,  30,   27,    9,    1;
  5: 11,  55,   77,   44,   11,    1;
  6: 13,  91,  182,  156,   65,   13,   1;
  7: 15, 140,  378,  450,  275,   90,  15,   1;
  8: 17, 204,  714, 1122,  935,  442, 119,  17,  1;
  9: 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) - P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 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 >= 1, 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-ad1929e911eec993
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-0bb8ed02ac535b32, e-19ca7a5e6e7a3cfc, e-ad1929e911eec993
  • Real-rooted. Every row P_n, n >= 1, 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-0bb8ed02ac535b32
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-0bb8ed02ac535b32, e-19ca7a5e6e7a3cfc
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-0bb8ed02ac535b32, e-19ca7a5e6e7a3cfc
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-19ca7a5e6e7a3cfc
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1, 1] has a real-rooted image
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-e1219184df277b70
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-4, 0]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-4, 0]
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-e1219184df277b70
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 0] to roots in [-4, 0]: every PF input with all roots in [-1, 0] has a PF image; no larger [-R, 0] works
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-e1219184df277b70

Proof chains

4 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 14 proof steps

Same fingerprint (4)

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