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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	413403	9748888	101767104	619284480	2433191936	6451200000	11692670976	14310965248	11324620800	5234491392	1073741824							53252096051
9	1542841	39946384	463187264	3174999552	14296821760	44412698624	97145323520	149661155328	159199002624	111400714240	46170898432	8589934592						634556225161
10	5757961	162379376	2070553024	15786211840	80154091520	285574168576	732190212096	1361496244224	1822760632320	1713960386560	1074815565824	403726925824	68719476736					7561422605881
11	21489003	655634808	9118059840	76534272000	432609239040	1739116707840	5116208873472	11153845518336	18023813480448	21326525890560	17964774457344	10204842295296	3504693313536	549755813888				90102515045411
12	80198051	2632071880	39646764800	363295354880	2262557040640	10131766575104	33645578944512	84183556816896	159503257436160	227782651019264	241396489388032	184133837914112	95588792139776	30236569763840	4398046511104			1073668757939051
13	299303201	10514237120	170525574400	1693821265920	11523981762560	56888405917696	210520239505408	594745013305344	1293457670799360	2165830137675776	2769882391248896	2657502424465408	1851921178558464	885106860359680	259484744155136	35184372088832		12793922580223201
14	1117014753	41819302208	726569429760	7776194304000	57383940136960	309613711196160	1263542626418688	3978958412447744	9772287532204032	18783199266078720	28164774177013760	32604934989938688	28572115317817344	18325560300142592	8114395812986880	2216615441596416	281474976710656	152453402204739361

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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