Real-rooted atlas

A114655 → D+1

nonnegative coefficients

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

Recurrence
\(\left(6 - 6\,x + x^{2} + 3\,n - 4\,n x + n x^{2} - 2\,n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(8 - 20\,x + 17\,x^{2} - 3\,x^{3} + 6\,n + 7\,n x - 19\,n x^{2} + 3\,n x^{3} - 6\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(10\,x - 21\,x^{2} + 12\,x^{3} - 2\,x^{4} - 6\,n x + 11\,n x^{2} - 4\,n x^{3} + 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
122
2628
32016238
46496302192
51924802804821002
6544208020006407025336
714728064117606160126096228814
838402867259584470401568022401262157184
99728952322688003010561505283494436961602864146
102406429952011059201677312118540841126470560576019824780008
115836890112042240008363520798336039029769979201320008580240226572086
12139264261324815178240381004804739328031223808112210562207040232320123202862148321344
13327680734822451836928161075200253693440217451520105101568289935364530240388960171603362830764794
1476185620127744169594880639614976124591667213506155528480609283140966406870864087447366246242329639024666890936
15175308853903360534876160240824729656899082247622078464604636032029053939208519871361515754241603201696824030940448226283115038

Computed in your browser ?

Technical details
Node id
n-7f2a74b39253adec · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-48b123f09be537c0

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     2;
   2:     6,      2;
   3:    20,     16,       2;
   4:    64,     96,      30,       2;
   5:   192,    480,     280,      48,       2;
   6:   544,   2080,    2000,     640,      70,      2;
   7:  1472,   8064,   11760,    6160,    1260,     96,     2;
   8:  3840,  28672,   59584,   47040,   15680,   2240,   126,    2;
   9:  9728,  95232,  268800,  301056,  150528,  34944,  3696,  160,   2;
  10: 24064, 299520, 1105920, 1677312, 1185408, 411264, 70560, 5760, 198, 2;
%F Formula
The row polynomials satisfy (n^3-2*n^2+3*n+6+(2*n^2-4*n-6)*x+(n+1)*x^2)*P(n,x) = (2*n^3+6*n+8+(3*n^3-6*n^2+7*n-20)*x+(6*n^2-19*n+17)*x^2+(3*n-3)*x^3)*P(n-1,x) + ((4*n^2-6*n+10)*x-(2*n^2-11*n+21)*x^2-(4*n-12)*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 6*n + 8 + (3*n^3 - 6*n^2 + 7*n - 20)*x + (6*n^2 - 19*n + 17)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + ((4*n^2 - 6*n + 10)*x - (2*n^2 - 11*n + 21)*x^2 - (4*n - 12)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + 3*n + 6 + (2*n^2 - 4*n - 6)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n^3 + 6*n + 8 + (3*n^3 - 6*n^2 + 7*n - 20)*x + (6*n^2 - 19*n + 17)*x^2 + (3*n - 3)*x^3)*v[i - 1] + ((4*n^2 - 6*n + 10)*x - (2*n^2 - 11*n + 21)*x^2 - (4*n - 12)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + 3*n + 6 + (2*n^2 - 4*n - 6)*x + (n + 1)*x^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 >= 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-c818d14e04074107
  • 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-c818d14e04074107
  • 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-c818d14e04074107
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-c818d14e04074107

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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.