Real-rooted atlas

A059473: Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

A059473 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A059473
\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2410418
383232880
416881488816356
532224536536224321584
664544169624401696544647048
712812804928934493444928128012831360

Show all 15 stored rows

Generated b-files: A059473, A059473 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(2\,x^{2} + 2\,n - n x - 2\,n x^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(12\,x^{2} + 6\,x^{3} - 2\,n x - 8\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 8\,x^{3} + 2\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-1bffaa6198344f93 · revision 2 · JSON
Definition source
oeis_catalog:A059473
derived:reversal (equal by e-f417b7c94fe88fc7)
Fingerprint
f6-324a05be44062cc1

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: yes
  • γ-positive (gamma_positive): yes
  • 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,    2;
  2:   4,   10,     4;
  3:   8,   32,    32,      8;
  4:  16,   88,   148,     88,     16;
  5:  32,  224,   536,    536,    224,     32;
  6:  64,  544,  1696,   2440,   1696,    544,     64;
  7: 128, 1280,  4928,   9344,   9344,   4928,   1280,   128;
  8: 256, 2944, 13504,  31936,  42256,  31936,  13504,  2944,  256;
  9: 512, 6656, 35456, 100736, 167072, 167072, 100736, 35456, 6656, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] + 2*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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] + 2*x*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.

A059473: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    2;
  2:   4,   10,     4;
  3:   8,   32,    32,      8;
  4:  16,   88,   148,     88,     16;
  5:  32,  224,   536,    536,    224,     32;
  6:  64,  544,  1696,   2440,   1696,    544,     64;
  7: 128, 1280,  4928,   9344,   9344,   4928,   1280,   128;
  8: 256, 2944, 13504,  31936,  42256,  31936,  13504,  2944,  256;
  9: 512, 6656, 35456, 100736, 167072, 167072, 100736, 35456, 6656, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] + 2*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] + 2*x*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.

A059473: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    2;
  2:   4,   10,     4;
  3:   8,   32,    32,      8;
  4:  16,   88,   148,     88,     16;
  5:  32,  224,   536,    536,    224,     32;
  6:  64,  544,  1696,   2440,   1696,    544,     64;
  7: 128, 1280,  4928,   9344,   9344,   4928,   1280,   128;
  8: 256, 2944, 13504,  31936,  42256,  31936,  13504,  2944,  256;
  9: 512, 6656, 35456, 100736, 167072, 167072, 100736, 35456, 6656, 512;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] + 2*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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] + 2*x*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.

Proved properties

  • 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-0ee228386582da48
  • 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-0ee228386582da48, e-c68c82cb1005adf9
  • 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-0ee228386582da48, e-c68c82cb1005adf9
  • 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-0ee228386582da48
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(3), -1 + sqrt(3)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1 - sqrt(3), -1 + sqrt(3)] 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-bab1139f7615a265
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(3), -1 + sqrt(3)] to roots in [-2 - sqrt(3), -2 + sqrt(3)]: every real polynomial with all roots in E = [-1 - sqrt(3), -1 + sqrt(3)] has an image with all roots in I = [-2 - sqrt(3), -2 + sqrt(3)]
    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-bab1139f7615a265
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1 - sqrt(3), 0] to roots in [-2 - sqrt(3), -2 + sqrt(3)]: every PF input with all roots in [-1 - sqrt(3), 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-bab1139f7615a265
  • 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-c68c82cb1005adf9

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 15 proof steps

No candidate equalities or related sequences.

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003