Real-rooted atlas

A391403: Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).

A011973 → ⊙ C(n,k)

A391403 nonnegative coefficients

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

Recurrence from OEIS A391403
\(\left(n\right)\,P_{n}(x) = \left(-1 + 2\,n\right)\,P_{n-1}(x) + \left(1 - 4\,x - n + 4\,n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2123
3167
4112619
51203051
61309020141
7142210140393

Show all 15 stored rows

Generated b-files: A002426 (row sums), A391403 ?

Computed in your browser ?

Other recurrences

Image under θ+1 · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(-4\,x + n + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Γ-polynomial · proved equal for all n (proof step)

\(\left(-n\right)\,P_{n}(x) = \left(1 - 2\,n\right)\,P_{n-1}(x) + \left(-1 + 4\,x + n - 4\,n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-8bfdd58279be42d7 · revision 2 · JSON
Definition source
oeis_catalog:A391403
derived:theta_plus_one (equal by e-b8fc68bdee02c71c)
derived:gamma_polynomial (equal by e-4502bda335658401)
Fingerprint
f6-d272e4b2958d4a94

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: 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,  2;
  3: 1,  6;
  4: 1, 12,   6;
  5: 1, 20,  30;
  6: 1, 30,  90,   20;
  7: 1, 42, 210,  140;
  8: 1, 56, 420,  560,  70;
  9: 1, 72, 756, 1680, 630;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n-1)*P(n-1,x) - (n-1-(4*n-4)*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[Together[((2*n - 1)*P[n - 1, x] - (n - 1 - (4*n - 4)*x)*P[n - 2, x])/(n)]];
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] = ((2*n - 1)*v[i - 1] - (n - 1 - (4*n - 4)*x)*v[i - 2])/(n)); 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.

A391403: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,  2;
  3: 1,  6;
  4: 1, 12,   6;
  5: 1, 20,  30;
  6: 1, 30,  90,   20;
  7: 1, 42, 210,  140;
  8: 1, 56, 420,  560,  70;
  9: 1, 72, 756, 1680, 630;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n-1)*P(n-1,x) - (n-1-(4*n-4)*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[Together[((2*n - 1)*P[n - 1, x] - (n - 1 - (4*n - 4)*x)*P[n - 2, x])/(n)]];
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] = ((2*n - 1)*v[i - 1] - (n - 1 - (4*n - 4)*x)*v[i - 2])/(n)); 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.

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-14cbfe8df545ebf2, e-2126fae20f2e5116, e-e4032a9809c053c3
  • 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-14cbfe8df545ebf2, e-2126fae20f2e5116, e-e4032a9809c053c3
  • 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-14cbfe8df545ebf2, e-2126fae20f2e5116, e-e4032a9809c053c3

Proof chains

5 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 5 proof steps

Same fingerprint (2)

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
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023