Real-rooted atlas

A078812: Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).

A078812 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 A078812
\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(-1\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
33418
44106121
5520218155
66355636101144
775612612055121377
888425233022078141987

Show all 15 stored rows

Generated b-files: A078812, A172431 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-976cdb6e4a693ec8 · revision 1 · JSON
Definition source
oeis_catalog:A078812
Fingerprint
f6-064b28e6ad672ed5

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:  2,   1;
   3:  3,   4,   1;
   4:  4,  10,   6,    1;
   5:  5,  20,  21,    8,    1;
   6:  6,  35,  56,   36,   10,    1;
   7:  7,  56, 126,  120,   55,   12,   1;
   8:  8,  84, 252,  330,  220,   78,  14,   1;
   9:  9, 120, 462,  792,  715,  364, 105,  16,  1;
  10: 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 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) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 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] = 2 + 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.

A078812: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   4,   1;
  3:  4,  10,   6,    1;
  4:  5,  20,  21,    8,    1;
  5:  6,  35,  56,   36,   10,    1;
  6:  7,  56, 126,  120,   55,   12,   1;
  7:  8,  84, 252,  330,  220,   78,  14,   1;
  8:  9, 120, 462,  792,  715,  364, 105,  16,  1;
  9: 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 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) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 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] = 2 + 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.

A172431: reversed triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  4,   3;
   4: 1,  6,  10,   4;
   5: 1,  8,  21,  20,    5;
   6: 1, 10,  36,  56,   35,    6;
   7: 1, 12,  55, 120,  126,   56,    7;
   8: 1, 14,  78, 220,  330,  252,   84,   8;
   9: 1, 16, 105, 364,  715,  792,  462, 120,   9;
  10: 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10;
%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 >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 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]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 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 = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-1502106ad4631c34
  • 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-1502106ad4631c34, e-537ef91063de36e5, e-5aa1c8cf88fab58c
  • 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-537ef91063de36e5
  • 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-537ef91063de36e5, e-5aa1c8cf88fab58c
  • 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-537ef91063de36e5, e-5aa1c8cf88fab58c
  • 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-5aa1c8cf88fab58c
  • 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-5a12e548117a4094
  • 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-5a12e548117a4094
  • 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-5a12e548117a4094

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 (5)

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