A008459 → basis A013609

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

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

\(P_{0}(x) = 1\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	6	12	4						22
3	20	60	48	8					136
4	70	280	360	160	16				886
5	252	1260	2240	1680	480	32			5944
6	924	5544	12600	13440	6720	1344	64		40636
7	3432	24024	66528	92400	67200	24192	3584	128	281488

Show all 15 stored rows

Generated b-files: A069835 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-fe801e460afa1e3b · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-a6dfa0000f2b0fe9

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 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A069835: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     2,      2;
  2:     6,     12,       4;
  3:    20,     60,      48,       8;
  4:    70,    280,     360,     160,      16;
  5:   252,   1260,    2240,    1680,     480,      32;
  6:   924,   5544,   12600,   13440,    6720,    1344,      64;
  7:  3432,  24024,   66528,   92400,   67200,   24192,    3584,    128;
  8: 12870, 102960,  336336,  576576,  554400,  295680,   80640,   9216,   256;
  9: 48620, 437580, 1647360, 3363360, 4036032, 2882880, 1182720, 253440, 23040, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 4)*x^2*P[n - 2, x] + (4*n - 2 + (4*n - 2)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 4)*x^2*v[i - 2] + (4*n - 2 + (4*n - 2)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); 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 >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4579edc5f7a4d216
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n 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-4579edc5f7a4d216
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-4579edc5f7a4d216
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-4579edc5f7a4d216

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 A013609 e-4579edc5f7a4d216

Geometric basis map (elementary) polytool ?
- premise (sequence): Square the entries of Pascal's triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-c6b04c4560617585 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - preserver_image reversal, shift 0 e-1906182487ade7a7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square the entries of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-c6b04c4560617585 (expanded above)

Related sequences ?

Same fingerprint (1)

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

A069466 → basis A013609

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