Real-rooted atlas

A021009: Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

A123202 → x ↦ x/(x+1)

A021009

Proved for all n: all roots real what this means

Recurrence from OEIS A021009
\(P_{n}(x) = \left(-1 - x + n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
21-10
32-41-1
46-189-1-4
524-9672-161-15
6120-600600-20025-1-56
7720-43205400-2400450-361-185
85040-3528052920-294007350-88249-1-204

Show all 15 stored rows

Generated b-files: A009940 (row sums), A021009, A021010 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-fcc65ac266ed7c2c · revision 1 · JSON
Definition source
oeis_catalog:A021009
Fingerprint
f6-aab0bc1aef1db223

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: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      1,       -1;
   3:      2,       -4,       1;
   4:      6,      -18,       9,       -1;
   5:     24,      -96,      72,      -16,       1;
   6:    120,     -600,     600,     -200,      25,      -1;
   7:    720,    -4320,    5400,    -2400,     450,     -36,     1;
   8:   5040,   -35280,   52920,   -29400,    7350,    -882,    49,    -1;
   9:  40320,  -322560,  564480,  -376320,  117600,  -18816,  1568,   -64,  1;
  10: 362880, -3265920, 6531840, -5080320, 1905120, -381024, 42336, -2592, 81, -1;
%F Formula
The row polynomials satisfy P(n,x) = (n-1-x)*P(n-1,x) + x*d/dx 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] + x*D[P[n - 1, x], 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] + x*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A021009: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:      1;
  1:      1,       -1;
  2:      2,       -4,       1;
  3:      6,      -18,       9,       -1;
  4:     24,      -96,      72,      -16,       1;
  5:    120,     -600,     600,     -200,      25,      -1;
  6:    720,    -4320,    5400,    -2400,     450,     -36,     1;
  7:   5040,   -35280,   52920,   -29400,    7350,    -882,    49,    -1;
  8:  40320,  -322560,  564480,  -376320,  117600,  -18816,  1568,   -64,  1;
  9: 362880, -3265920, 6531840, -5080320, 1905120, -381024, 42336, -2592, 81, -1;
%F Formula
The row polynomials satisfy P(n,x) = (n-x)*P(n-1,x) + x*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 - x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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 - x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A021010: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0:  1;
  1: -1,   1;
  2:  1,  -4,     2;
  3: -1,   9,   -18,      6;
  4:  1, -16,    72,    -96,      24;
  5: -1,  25,  -200,    600,    -600,     120;
  6:  1, -36,   450,  -2400,    5400,   -4320,      720;
  7: -1,  49,  -882,   7350,  -29400,   52920,   -35280,    5040;
  8:  1, -64,  1568, -18816,  117600, -376320,   564480, -322560,    40320;
  9: -1,  81, -2592,  42336, -381024, 1905120, -5080320, 6531840, -3265920, 362880;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n-x)*P(n-1,x) + x*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 - x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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 - x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-95f26075ae445d0d

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 (1)

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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023