A123519 → basis A013611

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 11 + 56\,x + 64\,x^{2}\)

\(P_{1}(x) = 41 + 304\,x + 704\,x^{2} + 512\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	11	56	64								131
1	41	304	704	512							1561
2	153	1488	5184	7680	4096						18601
3	571	6872	31936	71680	77824	32768					221651
4	2131	30568	177536	534528	880640	753664	262144				2641211
5	7953	132448	922752	3486720	7720960	10027008	7077888	2097152			31472881
6	29681	562848	4573056	20794368	57896960	101122048	108265472	65011712	16777216		375033361
7	110771	2356392	21872256	116275200	390221824	857636864	1234960384	1124073472	587202560	134217728	4468927451

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-07c6a4edd51718c6 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-4309d584dc07719c

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A077417: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:      11,       56,        64;
  1:      41,      304,       704,        512;
  2:     153,     1488,      5184,       7680,        4096;
  3:     571,     6872,     31936,      71680,       77824,       32768;
  4:    2131,    30568,    177536,     534528,      880640,      753664,      262144;
  5:    7953,   132448,    922752,    3486720,     7720960,    10027008,     7077888,      2097152;
  6:   29681,   562848,   4573056,   20794368,    57896960,   101122048,   108265472,     65011712,     16777216;
  7:  110771,  2356392,  21872256,  116275200,   390221824,   857636864,  1234960384,   1124073472,    587202560,    134217728;
  8:  413403,  9748888, 101767104,  619284480,  2433191936,  6451200000, 11692670976,  14310965248,  11324620800,   5234491392,  1073741824;
  9: 1542841, 39946384, 463187264, 3174999552, 14296821760, 44412698624, 97145323520, 149661155328, 159199002624, 111400714240, 46170898432, 8589934592;

%F Formula

The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 11 + 56*x + 64*x^2, P(1,x) = 41 + 304*x + 704*x^2 + 512*x^3.

%t Mathematica

P[0, x_] := 11 + 56*x + 64*x^2;
P[1, x_] := 41 + 304*x + 704*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 11 + 56*x + 64*x^2; v[2] = 41 + 304*x + 704*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 8*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 - 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-ab96494d3bfaa8ca
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-ab96494d3bfaa8ca
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-ab96494d3bfaa8ca
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-ab96494d3bfaa8ca

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 A013611 e-ab96494d3bfaa8ca

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-2a4594def0262eef root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

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