Real-rooted atlas

A152198: Triangle read by rows, A007318 rows repeated.

A152198 A152815 palindromic γ-positive nonnegative coefficients only rational roots

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

Recurrence from OEIS A152198
\(P_{n}(x) = \left(1 + x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2112
3112
41214
51214
613318
713318

Show all 15 stored rows

Generated b-files: A016116 (row sums), A060546 (row sums), A152198, A152198 (reversed triangle), A163403 (row sums) ?

Computed in your browser ?

Other recurrences

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

\(P_{n}(x) = \left(1 - x + n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-4124f4242e7d4c67 · revision 2 · JSON
Definition source
oeis_catalog:A152198
derived:reversal (equal by e-ccc9c8e3089cb292)
Fingerprint
f6-6052c7b42ebfb2ec

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1;
  2: 1, 1;
  3: 1, 1;
  4: 1, 2, 1;
  5: 1, 2, 1;
  6: 1, 3, 3, 1;
  7: 1, 3, 3, 1;
  8: 1, 4, 6, 4, 1;
  9: 1, 4, 6, 4, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + 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.

A152198: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1, 1;
  3: 1, 1;
  4: 1, 2, 1;
  5: 1, 2, 1;
  6: 1, 3, 3, 1;
  7: 1, 3, 3, 1;
  8: 1, 4, 6, 4, 1;
  9: 1, 4, 6, 4, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + 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.

A152198: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1, 1;
  3: 1, 1;
  4: 1, 2, 1;
  5: 1, 2, 1;
  6: 1, 3, 3, 1;
  7: 1, 3, 3, 1;
  8: 1, 4, 6, 4, 1;
  9: 1, 4, 6, 4, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); 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-3ca43d0d97d5e532, e-78c582ff5f7b6a70, e-b274e5cc6dbb8c2d
  • 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-3ca43d0d97d5e532, e-78c582ff5f7b6a70, e-b274e5cc6dbb8c2d
  • 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-3ca43d0d97d5e532, e-78c582ff5f7b6a70, e-b274e5cc6dbb8c2d
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, -1]: every PF input has a real-rooted image with every root <= -c/b = -1
    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-843ee697646ea568, e-ccad66a240ec24f8
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, -1]: every PF input has a PF image (zero allowed)
    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-843ee697646ea568, e-ccad66a240ec24f8

Proof chains

6 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 12 proof steps

Same fingerprint (6)

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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10