Real-rooted atlas

A152198 → x ↦ x+1

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(-x + n + n x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2213
3213
44419
54419
68126127
78126127

Show all 15 stored rows

Generated b-files: A056449 (row sums), A108411 (row sums), A162436 (row sums), A188440 (reversed triangle) ?

Computed in your browser ?

Other recurrences

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

\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-8c8ff2e102336282 · revision 1 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-acd9532b042548d9)
Fingerprint
f6-98cc1557e904900a

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): 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,  1;
  3:  2,  1;
  4:  4,  4,  1;
  5:  4,  4,  1;
  6:  8, 12,  6, 1;
  7:  8, 12,  6, 1;
  8: 16, 32, 24, 8, 1;
  9: 16, 32, 24, 8, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-1,x) - (4+6*x+2*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[(n + (n - 1)*x)*P[n - 1, x] - (4 + 6*x + 2*x^2)*D[P[n - 1, x], 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 + (n - 1)*x)*v[i - 1] - (4 + 6*x + 2*x^2)*deriv(v[i - 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.

A188440: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1, 2;
  3: 1, 2;
  4: 1, 4,  4;
  5: 1, 4,  4;
  6: 1, 6, 12,  8;
  7: 1, 6, 12,  8;
  8: 1, 8, 24, 32, 16;
  9: 1, 8, 24, 32, 16;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+(n-1)*x)*P(n-1,x) - (4+6*x+2*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[(n + (n - 1)*x)*P[n - 1, x] - (4 + 6*x + 2*x^2)*D[P[n - 1, x], x]];
Table[Reverse[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 + (n - 1)*x)*v[i - 1] - (4 + 6*x + 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); 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-009b68c8336dc2e0
  • 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-009b68c8336dc2e0, e-228642d15a5ba9ae
  • 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-009b68c8336dc2e0, e-228642d15a5ba9ae
  • 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-228642d15a5ba9ae

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 (10)

Equal up to content, sign, reversal and a small index shift.