Real-rooted atlas

A271697: Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

A271697 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 A271697
\(P_{n}(x) = \left(-2\,x + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x) + \left(-2\,x + n x\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 0\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
20
3011
40112
501719
6012121144
70151161511265
80111381381311311854

Show all 15 stored rows

Generated b-files: A000166 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-b24bb505650436c3 · revision 1 · JSON
Definition source
oeis_catalog:A271697
Fingerprint
f6-d85204b0dfc31b00

Observed on the stored rows ?

  • Degrees: 0, -1, 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: 0;
   3: 0, 1;
   4: 0, 1,   1;
   5: 0, 1,   7,     1;
   6: 0, 1,  21,    21,     1;
   7: 0, 1,  51,   161,    51,     1;
   8: 0, 1, 113,   813,   813,   113,     1;
   9: 0, 1, 239,  3361,  7631,  3361,   239,   1;
  10: 0, 1, 493, 12421, 53833, 53833, 12421, 493, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n-2)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + (n-2)*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 0.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 0;
P[n_, x_] := P[n, x] = Expand[(n - 2)*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + (n - 2)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; for(n = 3, N, my(i = n); v[i] = (n - 2)*x*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + (n - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 3, 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-f3672e8db5e390e2
  • Coefficients of one sign. Every nonzero row, n >= 3: 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-f3672e8db5e390e2
  • Hadamard kernel (Malo). Rows P_n, n >= 3, 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-f3672e8db5e390e2
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 4
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f3672e8db5e390e2

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

Same fingerprint (1)

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