Real-rooted atlas

A123519 → basis A013609 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 55 + 60\,x + 16\,x^{2}\)
\(P_{1}(x) = 433 + 696\,x + 368\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0556016131
1433696368641561
2340972405712198425618601
32683970860742883865699841024221651
42113036669968720326044162342404812840962641211
51663585611032095699528284800428160013209602252801638431472881
6130973775486990410012886410395379267157760276459527081984103219265536375033361
7103115431485238420101094057612238609929487956484884776961670144003656908846530562621444468927451
88118260714239499180992834942413730696448124186513927675358208328294400095957811218343526420709376104857653252096051
96391493137366780593049537385153614833510828815332320128011058899353656797970432207718318085301141504899153920912261124194304634556225161
1050320119025314750947800899774700080155444557600018075073920001490329395200893456793600392406958080125313024000283770880004325376000398458880167772167561422605881
113961694590632682609999196836182754030415886328300032205245182387201904207573606413052173959168669231100723225668308828167273696460801480201338882048498073617280532486710886490102515045411
1231190355534792273080688202076725285619136158983490985472225933951717888232944349448192179692237824000105354776936448471785780674561605790361190440893142794247555619225609574757171274490839042684354561073668757939051
1324556114968769191639987270880696363684940864156288274206028824228810594462722748248526721024235626312642560015549148557803527962809014026243164501715189769679810854912022381267517440378650021068844251584921631943819264107374182412793922580223201
1419332988419667316086135511600966260744142991296151295331852602882540864549209344031444568102105088296634068804648962175901657500876812542728056274944570066707415040020360962401894405654870124134401197213241835521867867855257620253455155201363652116484294967296152453402204739361
Technical details
Node id
n-4014ecdeba69f089 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-4231459a8d699316

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         55,          60,          16;
  1:        433,         696,         368,           64;
  2:       3409,        7240,        5712,         1984,          256;
  3:      26839,       70860,       74288,        38656,         9984,         1024;
  4:     211303,      666996,      872032,       604416,       234240,        48128,        4096;
  5:    1663585,     6110320,     9569952,      8284800,      4281600,      1320960,      225280,       16384;
  6:   13097377,    54869904,   100128864,    103953792,     67157760,     27645952,     7081984,     1032192,      65536;
  7:  103115431,   485238420,  1010940576,   1223860992,    948795648,    488477696,   167014400,    36569088,    4653056,    262144;
  8:  811826071,  4239499180,  9928349424,  13730696448,  12418651392,   7675358208,  3282944000,   959578112,  183435264,  20709376,  1048576;
  9: 6391493137, 36678059304, 95373851536, 148335108288, 153323201280, 110588993536, 56797970432, 20771831808, 5301141504, 899153920, 91226112, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 55 + 60*x + 16*x^2, P(1,x) = 433 + 696*x + 368*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 55 + 60*x + 16*x^2;
P[1, x_] := 433 + 696*x + 368*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (8 + 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] = 55 + 60*x + 16*x^2; v[2] = 433 + 696*x + 368*x^2 + 64*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (8 + 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)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023