Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → D

nonnegative coefficients

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

Recurrence
\(\left(-1 + n\right)\,P_{n}(x) = \left(n - 2\,n x^{2} + n^{2} + 4\,n^{2} x + 3\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-n x - 2\,n x^{2} - n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 2\,x - 2\,n - 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(1 + 2\,x + x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2527
32320346
4109152574322
5544106267214452427
62876728266122524345619645
71611350416602673388487458047170236

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-0c89f20f82728361 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-53105ee0eb8f88ac

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 ?

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:      0;
  1:      1;
  2:      5,       2;
  3:     23,      20,       3;
  4:    109,     152,      57,       4;
  5:    544,    1062,     672,     144,       5;
  6:   2876,    7282,    6612,    2524,     345,      6;
  7:  16113,   50416,   60267,   33884,    8745,    804,     7;
  8:  95495,  357128,  532893,  400684,  156655,  29028,  1841,    8;
  9: 597155, 2604902, 4679364, 4437892, 2359135, 682842, 94178, 4160, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3-n^2-n+(2*n^3-4*n^2)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (n^2+(3*n^2-n)*x+(3*n^2-2*n)*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2-2*n+1+(2*n^2-4*n+2)*x)*P(n-1,x) - (n-1+(2*n-2)*x+(n-1)*x^2)*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 - n^2 - n + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (n^2 + (3*n^2 - n)*x + (3*n^2 - 2*n)*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] - (n - 1 + (2*n - 2)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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 - n^2 - n + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n^2 + (3*n^2 - n)*x + (3*n^2 - 2*n)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 2*n + 1 + (2*n^2 - 4*n + 2)*x)*v[i - 1] - (n - 1 + (2*n - 2)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-1c9f283a75acfd0f
  • 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-1c9f283a75acfd0f
  • 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-1c9f283a75acfd0f

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. [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.