Real-rooted atlas

basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

palindromic γ-positive only rational roots

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

Recurrence
\(P_{n}(x) = \left(x - n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
20-1-1
30022
4000-6-6
500002424
600000-120-120
7000000720720
80000000-5040-5040

Show all 15 stored rows

Generated b-files: A133942 (row sums) ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(x + x^{2} - n x - n x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-cf1765de6ac2bd4b · revision 2 · JSON
Definition source
derived:basis_map_coefficients_of:oeis_catalog:A021012
derived:mobius_x_over_x_plus_one (equal by e-54c20f28143b4645)
Fingerprint
f6-edf6954a4cd8adfc

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 -1)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): no
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, -1;
   3: 0,  0, 2;
   4: 0,  0, 0, -6;
   5: 0,  0, 0,  0, 24;
   6: 0,  0, 0,  0,  0, -120;
   7: 0,  0, 0,  0,  0,    0, 720;
   8: 0,  0, 0,  0,  0,    0,   0, -5040;
   9: 0,  0, 0,  0,  0,    0,   0,     0, 40320;
  10: 0,  0, 0,  0,  0,    0,   0,     0,     0, -362880;
%F Formula
The row polynomials satisfy P(n,x) = -(n-1)*x*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[-(n - 1)*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {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)*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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-51f2fe49bf45c5d7
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-51f2fe49bf45c5d7, e-bb5cb95fed86aa9c
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-51f2fe49bf45c5d7, e-bb5cb95fed86aa9c
  • Real-rooted. Every row P_n, n >= 1, 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-bb5cb95fed86aa9c

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

Used in 15 proof steps

Same fingerprint (75)

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