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

A144084 → x ↦ −x (shift -1)

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 ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-1	1							0
3	1	-4	2						-1
4	-1	9	-18	6					-4
5	1	-16	72	-96	24				-15
6	-1	25	-200	600	-600	120			-56
7	1	-36	450	-2400	5400	-4320	720		-185
8	-1	49	-882	7350	-29400	52920	-35280	5040	-204

Show all 15 stored rows

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 ?

A009940: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A021009: reversed triangle, 47 terms agree, 9 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A021010: triangle, 55 terms agree, 10 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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

preserver_image affine, shift -1 e-e31dcd4b09a7ff12

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-c98efb5092e3dc49 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 2 proof steps

e-1c58af07f6ac5587 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients of Laguerre polynomials L_n(x) (power…
e-1c58af07f6ac5587 obreschkoff_combination → obreschkoff_combination of Triangle of coefficients of Laguerre polynomials L_n(x) (power…

Related sequences ?

Same fingerprint (1)

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

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

References

[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