Real-rooted atlas

A008278 → D

nonnegative coefficients

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

Recurrence
\(\left(-2 + n\right)\,P_{n}(x) = \left(1 - 6\,x + 6\,x^{2} - n + 8\,n x - 11\,n x^{2} - 2\,n^{2} x + 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x^{2} + 2\,x^{3} + n x^{2} - 3\,n x^{3} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-3 + 8\,x + 2\,n - 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3325
4614323
51050454109
6151302701245544
7212801050120431562876
828532315068044830762716113

Show all 15 stored rows

Generated b-files: A278677 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-e154bfba3f057e18 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-f041ba2901346a21

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 -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  0;
   2:  1;
   3:  3,    2;
   4:  6,   14,     3;
   5: 10,   50,    45,     4;
   6: 15,  130,   270,   124,      5;
   7: 21,  280,  1050,  1204,    315,      6;
   8: 28,  532,  3150,  6804,   4830,    762,     7;
   9: 36,  924,  7938, 27804,  38850,  18150,  1785,    8;
  10: 45, 1500, 17640, 91308, 212625, 204630, 65310, 4088, 9;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(n-1+(2*n^2-8*n+6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) + ((n-1)*x^2+(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (2*n-3+(2*n^2-8*n+8)*x)*P(n-1,x) - (n-2)*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 1 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] + ((n - 1)*x^2 + (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (2*n - 3 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] - (n - 2)*x^2*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n - 1 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] + ((n - 1)*x^2 + (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (2*n - 3 + (2*n^2 - 8*n + 8)*x)*v[i - 1] - (n - 2)*x^2*deriv(v[i - 1], x))/(n - 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-0316e05e74194510
  • 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-0316e05e74194510
  • 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-0316e05e74194510
  • 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-0316e05e74194510

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.