Real-rooted atlas

A062196 → 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
\(\left(-4\,n x + 2\,n x^{2} + 2\,n^{2} x + n^{2} x^{2} + 2\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(-2\,x^{2} + 2\,x^{3} + 2\,n - 4\,n x + 2\,n x^{2} + 4\,n x^{3} + 5\,n^{2} + 2\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + 4\,n^{3} + 8\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(2\,x^{3} - 2\,x^{4} + 2\,n x + 2\,n x^{2} - 2\,n x^{3} - 2\,n x^{4} + 5\,n^{2} x - n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
2920635
316756010161
42520433014015714
53645512601050280213102
6498883780532027305042813299
76415759576205801764061748403656485
88126002142065520852604939212600132045238238
910040594356018018033264029106012196823760198055999362
1012160608217044220011018701363824859320273240420752860664173806
11144872314586099099032089205351346475675222651205662807078540047817369002
1216912180246246206206084234151823421621801780146100245450445110110011411454609172061300
1319616575398580403403020300280553963418588580077297220404890201216715520300281774507280105298200060
142252206462244074911204554550015303288029888258434845096024477453010306296025505480357739226754095201201231245315

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Technical details
Node id
n-cff3d698fea91c3e · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-511eb58d09e2ef39

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 ?

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:   4,    3;
  2:   9,   20,     6;
  3:  16,   75,    60,     10;
  4:  25,  204,   330,    140,     15;
  5:  36,  455,  1260,   1050,    280,     21;
  6:  49,  888,  3780,   5320,   2730,    504,     28;
  7:  64, 1575,  9576,  20580,  17640,   6174,    840,    36;
  8:  81, 2600, 21420,  65520,  85260,  49392,  12600,  1320,   45;
  9: 100, 4059, 43560, 180180, 332640, 291060, 121968, 23760, 1980, 55;
%F Formula
The row polynomials satisfy (n^4+2*n^3+(2*n^3+2*n^2-4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (n^4+4*n^3+5*n^2+2*n+(3*n^4+8*n^3+2*n^2-4*n)*x+(6*n^3+6*n^2+2*n-2)*x^2+(3*n^2+4*n+2)*x^3)*P(n-1,x) + ((2*n^3+5*n^2+2*n)*x-(2*n^3+n^2-2*n)*x^2-(4*n^2+2*n-2)*x^3-(2*n+2)*x^4)*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[((n^4 + 4*n^3 + 5*n^2 + 2*n + (3*n^4 + 8*n^3 + 2*n^2 - 4*n)*x + (6*n^3 + 6*n^2 + 2*n - 2)*x^2 + (3*n^2 + 4*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 5*n^2 + 2*n)*x - (2*n^3 + n^2 - 2*n)*x^2 - (4*n^2 + 2*n - 2)*x^3 - (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 + (2*n^3 + 2*n^2 - 4*n)*x + (n^2 + 2*n)*x^2)]];
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] = ((n^4 + 4*n^3 + 5*n^2 + 2*n + (3*n^4 + 8*n^3 + 2*n^2 - 4*n)*x + (6*n^3 + 6*n^2 + 2*n - 2)*x^2 + (3*n^2 + 4*n + 2)*x^3)*v[i - 1] + ((2*n^3 + 5*n^2 + 2*n)*x - (2*n^3 + n^2 - 2*n)*x^2 - (4*n^2 + 2*n - 2)*x^3 - (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 + (2*n^3 + 2*n^2 - 4*n)*x + (n^2 + 2*n)*x^2)); 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-2305188708ae9104
  • 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-2305188708ae9104
  • 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-2305188708ae9104
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-2305188708ae9104

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. [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.