Real-rooted atlas

A130534 → x ↦ x/(x+1)

nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
227615
36294624105
424146329326120945
512087425213604255672010395
67206084212443927140564222125040135135
7504048348197380444849598116479996212976403202027025

Show all 15 stored rows

Generated b-files: A059364 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-5051121a22eaf16b · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-774e45ab36bcae1f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): yes

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:      1;
  1:      1,       2;
  2:      2,       7,        6;
  3:      6,      29,       46,       24;
  4:     24,     146,      329,      326,       120;
  5:    120,     874,     2521,     3604,      2556,       720;
  6:    720,    6084,    21244,    39271,     40564,     22212,      5040;
  7:   5040,   48348,   197380,   444849,    598116,    479996,    212976,    40320;
  8:  40320,  432144,  2014172,  5335212,   8788569,   9223012,   6023772,  2239344,   362880;
  9: 362880, 4292496, 22448988, 68158628, 132449241, 170892798, 146444068, 80391816, 25659360, 3628800;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+(2*n+1)*x+(n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (2*n + 1)*x + (n + 1)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + (2*n + 1)*x + (n + 1)*x^2)*v[i - 1])/(1 + x)); 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.

A059364: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        7,        2;
   4:      24,       46,       29,         6;
   5:     120,      326,      329,       146,        24;
   6:     720,     2556,     3604,      2521,       874,       120;
   7:    5040,    22212,    40564,     39271,     21244,      6084,      720;
   8:   40320,   212976,   479996,    598116,    444849,    197380,    48348,     5040;
   9:  362880,  2239344,  6023772,   9223012,   8788569,   5335212,  2014172,   432144,   40320;
  10: 3628800, 25659360, 80391816, 146444068, 170892798, 132449241, 68158628, 22448988, 4292496, 362880;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (1+x)*P(n,x) = (n-1+(2*n-1)*x+n*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (2*n - 1)*x + n*x^2)*P[n - 1, x])/(1 + x)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n - 1 + (2*n - 1)*x + n*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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 [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-034d7e75529eb2bf
  • 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-034d7e75529eb2bf
  • 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-034d7e75529eb2bf

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 (2)

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