Real-rooted atlas

A130534 → D

nonnegative coefficients

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

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2325
31112326
45070304154
52744502556051044
617643248220570010568028
713068262642030778401610168769264

Show all 15 stored rows

Generated b-files: A001705 (row sums), A074246 ?

Computed in your browser ?

Technical details
Node id
n-a34e8f08502aec80 · revision 1 · JSON
Definition source
derived:derivative
Fingerprint
f6-792365468f8ca6c1

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:       3,       2;
  3:      11,      12,       3;
  4:      50,      70,      30,       4;
  5:     274,     450,     255,      60,      5;
  6:    1764,    3248,    2205,     700,    105,     6;
  7:   13068,   26264,   20307,    7840,   1610,   168,    7;
  8:  109584,  236248,  201852,   89796,  22680,  3276,  252,   8;
  9: 1026576, 2345400, 2171040, 1077300, 316365, 56700, 6090, 360, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(2*n-2)*x+x^2)*P(n-2,x) + (2*n-1+2*x)*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[-(n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*P[n - 2, x] + (2*n - 1 + 2*x)*P[n - 1, x]];
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^2 - 2*n + 1 + (2*n - 2)*x + x^2)*v[i - 2] + (2*n - 1 + 2*x)*v[i - 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.

A074246: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1:        1;
   2:        3,        2;
   3:       11,       12,        3;
   4:       50,       70,       30,        4;
   5:      274,      450,      255,       60,       5;
   6:     1764,     3248,     2205,      700,     105,      6;
   7:    13068,    26264,    20307,     7840,    1610,    168,      7;
   8:   109584,   236248,   201852,    89796,   22680,   3276,    252,     8;
   9:  1026576,  2345400,  2171040,  1077300,  316365,  56700,   6090,   360,   9;
  10: 10628640, 25507152, 25228500, 13667720, 4510275, 946638, 127050, 10560, 495, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(2*n-2)*x+x^2)*P(n-2,x) + (2*n-1+2*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*P[n - 2, x] + (2*n - 1 + 2*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 2*n + 1 + (2*n - 2)*x + x^2)*v[i - 2] + (2*n - 1 + 2*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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 >= 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-9a8da9628cbb79fd
  • 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-9a8da9628cbb79fd
  • 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-9a8da9628cbb79fd
  • 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-9a8da9628cbb79fd

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.