Real-rooted atlas

A113413 → D

nonnegative coefficients

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

Recurrence
\(\left(4 + 3\,x + x^{2} - 8\,n - 4\,n x + 5\,n^{2} - n^{2} x^{2} - n^{3} + n^{3} x\right)\,P_{n}(x) = \left(2 + 12\,x + 2\,x^{2} - 5\,n - 30\,n x - 18\,n x^{2} - 3\,n x^{3} + 4\,n^{2} + 21\,n^{2} x + 7\,n^{2} x^{2} - n^{3} - 3\,n^{3} x\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + x^{3} - 2\,n x - 13\,n x^{2} - 8\,n x^{3} - n x^{4} + n^{2} x + 6\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2426
3812323
4123624476
5167696405233
620132264200606682
7242045766803608471935

Show all 15 stored rows

Generated b-files: A054459 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-6998812e4619b843 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-e1fbdf0e40e7644a

Observed on the stored rows ?

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

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:  0;
  1:  1;
  2:  4,   2;
  3:  8,  12,    3;
  4: 12,  36,   24,    4;
  5: 16,  76,   96,   40,    5;
  6: 20, 132,  264,  200,   60,    6;
  7: 24, 204,  576,  680,  360,   84,   7;
  8: 28, 292, 1080, 1800, 1460,  588, 112,   8;
  9: 32, 396, 1824, 4008, 4560, 2772, 896, 144, 9;
%F Formula
The row polynomials satisfy -(n^3-5*n^2+8*n-4-(n^3-4*n+3)*x+(n^2-1)*x^2)*P(n,x) = -(n^3-4*n^2+5*n-2+(3*n^3-21*n^2+30*n-12)*x-(7*n^2-18*n+2)*x^2+3*n*x^3)*P(n-1,x) + ((n^2-2*n+1)*x+(6*n^2-13*n+6)*x^2+(n^2-8*n+1)*x^3-n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 5*n - 2 + (3*n^3 - 21*n^2 + 30*n - 12)*x - (7*n^2 - 18*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] + ((n^2 - 2*n + 1)*x + (6*n^2 - 13*n + 6)*x^2 + (n^2 - 8*n + 1)*x^3 - n*x^4)*D[P[n - 1, x], x])/(-n^3 + 5*n^2 - 8*n + 4 + (n^3 - 4*n + 3)*x - (n^2 - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 5*n - 2 + (3*n^3 - 21*n^2 + 30*n - 12)*x - (7*n^2 - 18*n + 2)*x^2 + 3*n*x^3)*v[i - 1] + ((n^2 - 2*n + 1)*x + (6*n^2 - 13*n + 6)*x^2 + (n^2 - 8*n + 1)*x^3 - n*x^4)*deriv(v[i - 1], x))/(-n^3 + 5*n^2 - 8*n + 4 + (n^3 - 4*n + 3)*x - (n^2 - 1)*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-cbf712b709a8ecc6
  • 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-cbf712b709a8ecc6
  • 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-cbf712b709a8ecc6
  • 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-cbf712b709a8ecc6

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.