Real-rooted atlas

A021010: Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).

A144084 → x ↦ −x (shift -1)

A021010

Proved for all n: all roots real, in [0, ∞) · alternating coefficient signs · interlacing (weak) what this means

Recurrence from OEIS A021010
\(P_{n}(x) = \left(-1 - 3\,x + 2\,n x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-110
31-42-1
4-19-186-4
51-1672-9624-15
6-125-200600-600120-56
71-36450-24005400-4320720-185
8-149-8827350-2940052920-352805040-204
91-641568-18816117600-376320564480-322560403206209
10-181-259242336-3810241905120-50803206531840-3265920362880112400
111-1004050-864001058400-762048031752000-7257600081648000-3628800036288001520271
12-1121-6050163350-261360025613280-153679680548856000-10977120001097712000-4390848003991680019165420
131-1448712-2904005880600-75271680614718720-31614105609879408000-1756339200015807052800-5748019200479001600237686449
14-1169-12168490776-12269400198764280-212015232014841066240-66784798080185513328000-296821324800242853811200-8095127040062270208002944654296
151-19616562-79497624048024-4809604806492966480-59364264960363606122880-14544244915203636061228800-52888163328003966612249600-12204960768008717829120036392001815

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

Computed in your browser ?

Technical details
Node id
n-3de13f8c869a1c34 · revision 1 · JSON
Definition source
oeis_catalog:A021010
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:  1,  -4,     2;
   4: -1,   9,   -18,      6;
   5:  1, -16,    72,    -96,      24;
   6: -1,  25,  -200,    600,    -600,     120;
   7:  1, -36,   450,  -2400,    5400,   -4320,      720;
   8: -1,  49,  -882,   7350,  -29400,   52920,   -35280,    5040;
   9:  1, -64,  1568, -18816,  117600, -376320,   564480, -322560,    40320;
  10: -1,  81, -2592,  42336, -381024, 1905120, -5080320, 6531840, -3265920, 362880;
%F Formula
The row polynomials satisfy P(n,x) = -(1-(2*n-3)*x)*P(n-1,x) - x^2*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[-(1 - (2*n - 3)*x)*P[n - 1, x] - x^2*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] = -(1 - (2*n - 3)*x)*v[i - 1] - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

A021009: reversed 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
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(1-(2*n-1)*x)*P(n-1,x) - 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[-(1 - (2*n - 1)*x)*P[n - 1, x] - x^2*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] = -(1 - (2*n - 1)*x)*v[i - 1] - x^2*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.

A021010: 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
The row polynomials satisfy P(n,x) = -(1-(2*n-1)*x)*P(n-1,x) - 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[-(1 - (2*n - 1)*x)*P[n - 1, x] - x^2*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] = -(1 - (2*n - 1)*x)*v[i - 1] - x^2*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.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-e31dcd4b09a7ff12
  • Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n(-x) 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-e31dcd4b09a7ff12
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-e31dcd4b09a7ff12

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

Used in 2 proof steps

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