Real-rooted atlas

A113214 → D+1

nonnegative coefficients

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

Recurrence
\(\left(2\,n - 9\,n x + n x^{2} + n x^{3} - 3\,n^{2} + 3\,n^{2} x + n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(16 - 36\,x - 2\,x^{2} + 6\,x^{3} + x^{4} - 28\,n + 21\,n x + 4\,n x^{2} - n x^{3} + 14\,n^{2} - 3\,n^{2} x - 2\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x - 22\,x^{2} + 6\,x^{3} + 7\,x^{4} + x^{5} - 12\,n x + 9\,n x^{2} + 7\,n x^{3} + n x^{4} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
23519
32107120
4010209140
504293311178
600216349131147
70066611668151272
80003615919290171495
90008127325295115191890
100000553415944291432111584
1100001021877110015981742312796
120000078650154715868062082514901
13000001234516172842239410572452718540
140000001051135354248783475135528529114805

Computed in your browser ?

Technical details
Node id
n-cbfa3e440ba864d3 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4b74653ef77df9d6

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 ?

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: 3,  1;
  2: 3,  5,  1;
  3: 2, 10,  7,  1;
  4: 0, 10, 20,  9,   1;
  5: 0,  4, 29, 33,  11,   1;
  6: 0,  0, 21, 63,  49,  13,   1;
  7: 0,  0,  6, 66, 116,  68,  15,   1;
  8: 0,  0,  0, 36, 159, 192,  90,  17,  1;
  9: 0,  0,  0,  8, 127, 325, 295, 115, 19, 1;
%F Formula
The row polynomials satisfy (n^3-3*n^2+2*n+(3*n^2-9*n)*x+(n^2+n)*x^2+n*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16+(3*n^2-21*n+36)*x+(2*n^2-4*n+2)*x^2+(n-6)*x^3-x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2+9*n-22)*x^2+(7*n+6)*x^3+(n+7)*x^4+x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 14*n^2 + 28*n - 16 + (3*n^2 - 21*n + 36)*x + (2*n^2 - 4*n + 2)*x^2 + (n - 6)*x^3 - x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 + 9*n - 22)*x^2 + (7*n + 6)*x^3 + (n + 7)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 2*n + (3*n^2 - 9*n)*x + (n^2 + n)*x^2 + n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 + (3*n^2 - 21*n + 36)*x + (2*n^2 - 4*n + 2)*x^2 + (n - 6)*x^3 - x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 + 9*n - 22)*x^2 + (7*n + 6)*x^3 + (n + 7)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 2*n + (3*n^2 - 9*n)*x + (n^2 + n)*x^2 + n*x^3)); 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-3009f31f72baba04
  • 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-3009f31f72baba04
  • 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-3009f31f72baba04

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

No candidate equalities or related sequences.

References

  1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.