Real-rooted atlas

A114656 → 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
\(\left(6 - 6\,x + x^{2} + 3\,n - 4\,n x + n x^{2} - 2\,n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(8 - 20\,x + 17\,x^{2} - 3\,x^{3} + 6\,n + 7\,n x - 19\,n x^{2} + 3\,n x^{3} - 6\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(10\,x - 21\,x^{2} + 12\,x^{3} - 2\,x^{4} - 6\,n x + 11\,n x^{2} - 4\,n x^{3} + 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3108119
4324815196
596240140241501
6272104010003203512668
773640325880308063048114407
819201433629792235207840112063178592
948644761613440015052875264174721848801432073
10120321497605529608386565927042056323528028809912390004
11291844505602112000418176039916801951488498960660004290120113286043
126963213066247589120190502402369664015611904561052811035201161606160143174160672
13163840367411225918464805376001268467201087257605255078414496768226512019448085801681415382397
1438092810063872847974403198074886229583366753077764240304641570483203435432043723683123121164819512333445468
15876544269516802674380801204123648284495411238110392323023180160145269696042599356875787712801600848412015470224113141557519

Computed in your browser ?

Technical details
Node id
n-fb038b43f3455c33 · 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:     1;
   2:     3,      1;
   3:    10,      8,      1;
   4:    32,     48,     15,      1;
   5:    96,    240,    140,     24,      1;
   6:   272,   1040,   1000,    320,     35,      1;
   7:   736,   4032,   5880,   3080,    630,     48,     1;
   8:  1920,  14336,  29792,  23520,   7840,   1120,    63,    1;
   9:  4864,  47616, 134400, 150528,  75264,  17472,  1848,   80,  1;
  10: 12032, 149760, 552960, 838656, 592704, 205632, 35280, 2880, 99, 1;
%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) = 1.
%t Mathematica
P[1, x_] := 1;
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] = 1; 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-54d5821bb68ac50c
  • 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-54d5821bb68ac50c
  • 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-54d5821bb68ac50c
  • 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-54d5821bb68ac50c

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023