Real-rooted atlas

A011973 → basis A007318 → x ↦ x+1

nonnegative coefficients

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

Recurrence
\(\left(n\right)\,P_{n}(x) = \left(-4 - 2\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-18 - 17\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2314
3527
4117119
52116340
6434112197
78594344217

Show all 15 stored rows

Generated b-files: A006130 (row sums), A249139 ?

Computed in your browser ?

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-fe5f0667f4ce56c9 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-c2621fb6b5bc9d56)
Fingerprint
f6-3d8013fbbe82cb22

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): 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:   1;
  1:   1;
  2:   3,   1;
  3:   5,   2;
  4:  11,   7,   1;
  5:  21,  16,   3;
  6:  43,  41,  12,  1;
  7:  85,  94,  34,  4;
  8: 171, 219,  99, 18, 1;
  9: 341, 492, 261, 60, 5;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n-4+(2*n-2)*x)*P(n-1,x) - (18+17*x+4*x^2)*d/dx 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[((5*n - 4 + (2*n - 2)*x)*P[n - 1, x] - (18 + 17*x + 4*x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((5*n - 4 + (2*n - 2)*x)*v[i - 1] - (18 + 17*x + 4*x^2)*deriv(v[i - 1], x))/(n)); 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.

A249139: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:   1;
  1:   3,    1;
  2:   5,    2;
  3:  11,    7,   1;
  4:  21,   16,   3;
  5:  43,   41,  12,   1;
  6:  85,   94,  34,   4;
  7: 171,  219,  99,  18,  1;
  8: 341,  492, 261,  60,  5;
  9: 683, 1101, 678, 195, 25, 1;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (5*n+1+2*n*x)*P(n-1,x) - (18+17*x+4*x^2)*d/dx 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[((5*n + 1 + 2*n*x)*P[n - 1, x] - (18 + 17*x + 4*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n + 1 + 2*n*x)*v[i - 1] - (18 + 17*x + 4*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, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6b0b885431c4352b
  • 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-6b0b885431c4352b, e-c1bb5e4df6e83fab
  • 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-6b0b885431c4352b, e-c1bb5e4df6e83fab
  • 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-c1bb5e4df6e83fab

Proof chains

3 incoming proof steps. 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