Real-rooted atlas

A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

A130123 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A130123
\(P_{n}(x) = \left(2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
20044
300088
400001616
5000003232
60000006464
70000000128128

Show all 15 stored rows

Generated b-files: A000079 (row sums), A130123 ?

Computed in your browser ?

Other recurrences

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

\(\left(1 + x\right)\,P_{n}(x) = \left(2\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-ae7910bbe46f8a41 · revision 2 · JSON
Definition source
oeis_catalog:A130123
derived:mobius_x_over_x_plus_one (equal by e-dfb7962499f04eaa)
Fingerprint
f6-edf6954a4cd8adfc

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): yes

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: 0, 2;
  2: 0, 0, 4;
  3: 0, 0, 0, 8;
  4: 0, 0, 0, 0, 16;
  5: 0, 0, 0, 0,  0, 32;
  6: 0, 0, 0, 0,  0,  0, 64;
  7: 0, 0, 0, 0,  0,  0,  0, 128;
  8: 0, 0, 0, 0,  0,  0,  0,   0, 256;
  9: 0, 0, 0, 0,  0,  0,  0,   0,   0, 512;
%F Formula
The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = 2*x*v[i - 1]); 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.

A130123: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 0, 4;
  3: 0, 0, 0, 8;
  4: 0, 0, 0, 0, 16;
  5: 0, 0, 0, 0,  0, 32;
  6: 0, 0, 0, 0,  0,  0, 64;
  7: 0, 0, 0, 0,  0,  0,  0, 128;
  8: 0, 0, 0, 0,  0,  0,  0,   0, 256;
  9: 0, 0, 0, 0,  0,  0,  0,   0,   0, 512;
%F Formula
The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = 2*x*v[i - 1]); 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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-7963c2f0e7b86ba8
  • 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-37a63c273567eea5, e-7963c2f0e7b86ba8
  • 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-37a63c273567eea5, e-7963c2f0e7b86ba8
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
    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-764dc90b5c153977
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: an input root r <= 0 gives the root (r - a)/d <= 0; a larger root gives a positive root
    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-764dc90b5c153977
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-37a63c273567eea5

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 (75)

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