A123202: Triangle of coefficients of n!(1 - x)^nL_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.

Proved for all n: all roots real, in (0, 1) · alternating coefficient signs · interlacing what this means

Recurrence from OEIS A123202

\(P_{n}(x) = \left(-3 + 2\,x + 2\,n - 2\,n x\right)\,P_{n-1}(x) + \left(-4 + 8\,x - 4\,x^{2} + 4\,n - 8\,n x + 4\,n x^{2} - n^{2} + 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x)\)

\(P_{1}(x) = 1\)

\(P_{2}(x) = 1 - 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	-2							-1
3	2	-8	7						1
4	6	-36	63	-34					-1
5	24	-192	504	-544	209				1
6	120	-1200	4200	-6800	5225	-1546			-1
7	720	-8640	37800	-81600	94050	-55656	13327		1
8	5040	-70560	370440	-999600	1536150	-1363572	653023	-130922	-1

Show all 15 stored rows

Generated b-files: A123202 ?

Computed in your browser ?

Technical details

Node id: n-db839974123087ee · revision 1 · JSON
Definition source: oeis_catalog:A123202
Fingerprint: f6-e923cab30b4895b3

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 ?

A123202: triangle, 40 terms agree, 8 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,       -2;
   3:      2,       -8,        7;
   4:      6,      -36,       63,        -34;
   5:     24,     -192,      504,       -544,       209;
   6:    120,    -1200,     4200,      -6800,      5225,      -1546;
   7:    720,    -8640,    37800,     -81600,     94050,     -55656,     13327;
   8:   5040,   -70560,   370440,    -999600,   1536150,   -1363572,    653023,    -130922;
   9:  40320,  -645120,  3951360,  -12794880,  24578400,  -29089536,  20896736,   -8379008,   1441729;
  10: 362880, -6531840, 45722880, -172730880, 398170080, -589063104, 564211872, -339349824, 116780049, -17572114;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-3-(2*n-2)*x)*P(n-1,x) - (n^2-4*n+4-(2*n^2-8*n+8)*x+(n^2-4*n+4)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 - 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 - 2*x;
P[n_, x_] := P[n, x] = Expand[(2*n - 3 - (2*n - 2)*x)*P[n - 1, x] - (n^2 - 4*n + 4 - (2*n^2 - 8*n + 8)*x + (n^2 - 4*n + 4)*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 - 2*x; for(n = 3, N, my(i = n); v[i] = (2*n - 3 - (2*n - 2)*x)*v[i - 1] - (n^2 - 4*n + 4 - (2*n^2 - 8*n + 8)*x + (n^2 - 4*n + 4)*x^2)*v[i - 2]); 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.

A123202: triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0:      1;
  1:      1,       -2;
  2:      2,       -8,        7;
  3:      6,      -36,       63,        -34;
  4:     24,     -192,      504,       -544,       209;
  5:    120,    -1200,     4200,      -6800,      5225,      -1546;
  6:    720,    -8640,    37800,     -81600,     94050,     -55656,     13327;
  7:   5040,   -70560,   370440,    -999600,   1536150,   -1363572,    653023,    -130922;
  8:  40320,  -645120,  3951360,  -12794880,  24578400,  -29089536,  20896736,   -8379008,   1441729;
  9: 362880, -6531840, 45722880, -172730880, 398170080, -589063104, 564211872, -339349824, 116780049, -17572114;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-1-2*n*x)*P(n-1,x) - (n^2-2*n+1-(2*n^2-4*n+2)*x+(n^2-2*n+1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 - 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 - 2*x;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 - 2*n*x)*P[n - 1, x] - (n^2 - 2*n + 1 - (2*n^2 - 4*n + 2)*x + (n^2 - 2*n + 1)*x^2)*P[n - 2, 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; v[2] = 1 - 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 1 - 2*n*x)*v[i - 1] - (n^2 - 2*n + 1 - (2*n^2 - 4*n + 2)*x + (n^2 - 2*n + 1)*x^2)*v[i - 2]); 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 (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-6f81123042c063ba
Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 1)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-ddb7ec33b20a95f5
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-ddb7ec33b20a95f5
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-ddb7ec33b20a95f5

Proof chains

2 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

theorem_check three_term_recurrence e-6f81123042c063ba root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
theorem_check three_term_recurrence e-ddb7ec33b20a95f5 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 1 proof step

e-95f26075ae445d0d preserver_image (mobius, shift 0) → Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

Related sequences ?

No candidate equalities or related sequences.

References

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