Real-rooted atlas

A173018 → 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
\(\left(-4 + 2\,x + n\right)\,P_{n}(x) = \left(6 - 31\,x + 66\,x^{2} - 41\,x^{3} + 6\,x^{4} - 2\,n + 15\,n x - 44\,n x^{2} + 22\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 11\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(3\,x - 17\,x^{2} + 25\,x^{3} - 13\,x^{4} + 2\,x^{5} - n x + 8\,n x^{2} - 10\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-11 + 34\,x - 19\,x^{2} + 2\,x^{3} + 3\,n - 15\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 6\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
211
3213
456112
5123314160
627158144301360
75866112085306212520
81212502843971801791126120160

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-6e70ca4fb576e2d6 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d762c6db5469cb8f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:   1;
   3:   2,     1;
   4:   5,     6,      1;
   5:  12,    33,     14,      1;
   6:  27,   158,    144,     30,      1;
   7:  58,   661,   1208,    530,     62,      1;
   8: 121,  2502,   8439,   7180,   1791,    126,     1;
   9: 248,  8833,  51150,  78095,  37084,   5775,   254,   1;
  10: 503, 29718, 279310, 712994, 597360, 175882, 18122, 510, 1;
%F Formula
The row polynomials satisfy (n-4+2*x)*P(n,x) = -(2*n-6+(2*n^2-15*n+31)*x+(n^3-11*n^2+44*n-66)*x^2+(3*n^2-22*n+41)*x^3+(2*n-6)*x^4)*P(n-2,x) - ((n-3)*x+(n^2-8*n+17)*x^2-(n^2-10*n+25)*x^3-(3*n-13)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-11+(2*n^2-15*n+34)*x+(5*n-19)*x^2+2*x^3)*P(n-1,x) + ((n-4)*x-(n-6)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 6 + (2*n^2 - 15*n + 31)*x + (n^3 - 11*n^2 + 44*n - 66)*x^2 + (3*n^2 - 22*n + 41)*x^3 + (2*n - 6)*x^4)*P[n - 2, x] - ((n - 3)*x + (n^2 - 8*n + 17)*x^2 - (n^2 - 10*n + 25)*x^3 - (3*n - 13)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 11 + (2*n^2 - 15*n + 34)*x + (5*n - 19)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 4)*x - (n - 6)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 4 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n - 6 + (2*n^2 - 15*n + 31)*x + (n^3 - 11*n^2 + 44*n - 66)*x^2 + (3*n^2 - 22*n + 41)*x^3 + (2*n - 6)*x^4)*v[i - 2] - ((n - 3)*x + (n^2 - 8*n + 17)*x^2 - (n^2 - 10*n + 25)*x^3 - (3*n - 13)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 11 + (2*n^2 - 15*n + 34)*x + (5*n - 19)*x^2 + 2*x^3)*v[i - 1] + ((n - 4)*x - (n - 6)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 4 + 2*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-0053277d0044178d
  • 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-0053277d0044178d
  • 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-0053277d0044178d
  • 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-0053277d0044178d

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.