Real-rooted atlas

A334218: Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.

A334218 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A334218
\(\left(1 - \frac{1}{2}\,n\right)\,P_{n}(x) = \left(-x + \frac{3}{2}\,n x - \frac{1}{2}\,n^{2} x\right)\,P_{n-1}(x) + \left(\frac{1}{2}\,x - \frac{1}{2}\,x^{2} - \frac{1}{2}\,n x + \frac{1}{2}\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3022
40336
50416424
60555555120
7061563961566720
8073992114211439975040

Show all 15 stored rows

Generated b-files: A000142 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-3ffff24fcb4f40a0 · revision 1 · JSON
Definition source
oeis_catalog:A334218
Fingerprint
f6-605cb6decd8257f5

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1;
   3: 0, 2;
   4: 0, 3,    3;
   5: 0, 4,   16,     4;
   6: 0, 5,   55,    55,      5;
   7: 0, 6,  156,   396,    156,      6;
   8: 0, 7,  399,  2114,   2114,    399,     7;
   9: 0, 8,  960,  9528,  19328,   9528,   960,    8;
  10: 0, 9, 2223, 38637, 140571, 140571, 38637, 2223, 9;
%F Formula
The row polynomials satisfy -((1/2)*n-1)*P(n,x) = -((1/2)*n^2-(3/2)*n+1)*x*P(n-1,x) - (((1/2)*n-1/2)*x-((1/2)*n-1/2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((1/2)*n^2 - (3/2)*n + 1)*x*P[n - 1, x] - (((1/2)*n - 1/2)*x - ((1/2)*n - 1/2)*x^2)*D[P[n - 1, x], x])/(-(1/2)*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-((1/2)*n^2 - (3/2)*n + 1)*x*v[i - 1] - (((1/2)*n - 1/2)*x - ((1/2)*n - 1/2)*x^2)*deriv(v[i - 1], x))/(-(1/2)*n + 1)); 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.

Proved properties

  • 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-9baf8deff42ffdae
  • 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-9baf8deff42ffdae
  • 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-9baf8deff42ffdae
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-9baf8deff42ffdae

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 7 proof steps

Same fingerprint (6)

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

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