Real-rooted atlas

Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.

A166341 (tail) 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 A166341
\(P_{n}(x) = \left(1 + x + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 10\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1110112
212323148
3150138501240
4110561461410511440
51216236749122367216110080
61439839731483314838397439180640
7188628264176314314830176314282648861725760
811781918809031042632034263203490310491880178117257600

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-6a8ffbe72ba42678 · revision 1 · JSON
Definition source
oeis_catalog:A166341
Fingerprint
f6-9274366772724b5b

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 ?

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,   10,      1;
   2: 1,   23,     23,        1;
   3: 1,   50,    138,       50,         1;
   4: 1,  105,    614,      614,       105,         1;
   5: 1,  216,   2367,     4912,      2367,       216,         1;
   6: 1,  439,   8397,    31483,     31483,      8397,       439,         1;
   7: 1,  886,  28264,   176314,    314830,    176314,     28264,       886,        1;
   8: 1, 1781,  91880,   903104,   2632034,   2632034,    903104,     91880,     1781,      1;
   9: 1, 3572, 291669,  4347456,  19481898,  31584408,  19481898,   4347456,   291669,   3572,    1;
  10: 1, 7155, 910727, 20014845, 132189138, 325879734, 325879734, 132189138, 20014845, 910727, 7155, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 10*x + x^2.
%t Mathematica
P[1, x_] := 1 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 10*x + x^2; for(n = 2, N, my(i = n); v[i] = (1 + (n + 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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-b02efd7745c46534
  • 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-2e1314120c3b22a7, e-b02efd7745c46534
  • 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-2e1314120c3b22a7, e-b02efd7745c46534
  • 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-b02efd7745c46534
  • 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-2e1314120c3b22a7

Proof chains

2 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 3 proof steps

No candidate equalities or related sequences.

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