Real-rooted atlas

A207606 → 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
\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(3 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 13 + 44\,x + 32\,x^{2}\)
\(P_{1}(x) = 34 + 176\,x + 272\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
013443289
134176272128610
289620148814725124181
323320406672102407424204828657
46106432266885593662720358408192196418
515971969699120264320404480356352167936327681346269
641815904434945611335042208000265113619210247700481310729227465
710946174160118542445340161075353616429056161996809961472347340852428863245986
82865750722039034561721024048188672896501761123942409391308850135040154664962097152433494437
975025146212812553824627105282026534404452761606795837447213547525225840642464153606815744083886082971215073
101964184179264396065282211366408106137602056792064370746163247884861444403036160281411584011880366082977955843355443220365011074
115142291186133612298281675912550431137344008967555072186699694082847395020831840468992258080768001475241574456371445761291845632134217728139583862445
12134626933461660376787264254817113611567091456373008107528817266688015531324211220501417164820197199052814630151782475623301120263905607685570035712536870912956722026041
1335245789390872011412256168392536960417802245121492032430083950512742407901564436481204455014400140016458137612320400998408064388300803803730411521221381324802389075558421474836486557470319842
14922746526236281234225244642719434220814734372992057742981632016937941278723795361185792656897664614488163418112009150477107200727185358848043404838830081882336526336559687925760102005473280858993459244945570212853

Computed in your browser ?

Technical details
Node id
n-b85d9f7216f04287 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-d4f5100b4a0a6575

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 ?

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:    13,      44,       32;
  1:    34,     176,      272,      128;
  2:    89,     620,     1488,     1472,       512;
  3:   233,    2040,     6672,    10240,      7424,      2048;
  4:   610,    6432,    26688,    55936,     62720,     35840,      8192;
  5:  1597,   19696,    99120,   264320,    404480,    356352,    167936,     32768;
  6:  4181,   59044,   349456,  1133504,   2208000,   2651136,   1921024,    770048,    131072;
  7: 10946,  174160,  1185424,  4534016,  10753536,  16429056,  16199680,   9961472,   3473408,    524288;
  8: 28657,  507220,  3903456, 17210240,  48188672,  89650176, 112394240,  93913088,  50135040,  15466496,  2097152;
  9: 75025, 1462128, 12553824, 62710528, 202653440, 445276160, 679583744, 721354752, 522584064, 246415360, 68157440, 8388608;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 44*x + 32*x^2, P(1,x) = 34 + 176*x + 272*x^2 + 128*x^3.
%t Mathematica
P[0, x_] := 13 + 44*x + 32*x^2;
P[1, x_] := 34 + 176*x + 272*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 4*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] = 13 + 44*x + 32*x^2; v[2] = 34 + 176*x + 272*x^2 + 128*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + 4*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-b12bbff1e34cdc3a
  • 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-b12bbff1e34cdc3a
  • 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-b12bbff1e34cdc3a
  • 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-b12bbff1e34cdc3a

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

No candidate equalities or related sequences.

References

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