A021012: Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

basis-map coefficients of A021012 → basis A130595

A021012 only rational roots

Proved for all n: all roots real what this means

Recurrence from OEIS A021012

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	-1							0
3	2	-4	2						0
4	6	-18	18	-6					0
5	24	-96	144	-96	24				0
6	120	-600	1200	-1200	600	-120			0
7	720	-4320	10800	-14400	10800	-4320	720		0
8	5040	-35280	105840	-176400	176400	-105840	35280	-5040	0

Show all 15 stored rows

Generated b-files: A021012 ?

Computed in your browser ?

Technical details

Node id: n-95a5652eb8c6128a · revision 1 · JSON
Definition source: oeis_catalog:A021012
Fingerprint: f6-85f6fce8b02c5e6d

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): yes

OEIS matches ?

A021012: 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 ?

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,        2;
   4:      6,      -18,       18,        -6;
   5:     24,      -96,      144,       -96,       24;
   6:    120,     -600,     1200,     -1200,      600,      -120;
   7:    720,    -4320,    10800,    -14400,    10800,     -4320,      720;
   8:   5040,   -35280,   105840,   -176400,   176400,   -105840,    35280,     -5040;
   9:  40320,  -322560,  1128960,  -2257920,  2822400,  -2257920,  1128960,   -322560,   40320;
  10: 362880, -3265920, 13063680, -30481920, 45722880, -45722880, 30481920, -13063680, 3265920, -362880;

%F Formula

The row polynomials satisfy P(n,x) = (n-1-(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 - (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 - (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.

A021012: triangle, offset 0, from the row P₁

%e Example

Triangle begins:
  0:      1;
  1:      1,       -1;
  2:      2,       -4,        2;
  3:      6,      -18,       18,        -6;
  4:     24,      -96,      144,       -96,       24;
  5:    120,     -600,     1200,     -1200,      600,      -120;
  6:    720,    -4320,    10800,    -14400,    10800,     -4320,      720;
  7:   5040,   -35280,   105840,   -176400,   176400,   -105840,    35280,     -5040;
  8:  40320,  -322560,  1128960,  -2257920,  2822400,  -2257920,  1128960,   -322560,   40320;
  9: 362880, -3265920, 13063680, -30481920, 45722880, -45722880, 30481920, -13063680, 3265920, -362880;

%F Formula

The row polynomials satisfy P(n,x) = (n-n*x)*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 - n*x)*P[n - 1, 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 - n*x)*v[i - 1]); 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-679c0a5cb2a19fa4

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

basis_map A130595 e-679c0a5cb2a19fa4

Geometric basis map (elementary) polytool ?
- premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-51f2fe49bf45c5d7 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - preserver_image mobius, shift 0 e-bb5cb95fed86aa9c
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers root_domain e-51f2fe49bf45c5d7 (expanded above)

Related sequences ?

Same fingerprint (2)

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