Real-rooted atlas

A123519 → basis A007318 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 29 + 88\,x + 64\,x^{2}\)
\(P_{1}(x) = 169 + 752\,x + 1088\,x^{2} + 512\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0298864181
116975210885122521
298557761248011776409635113
357414178412000016998411878432768489061
43346129085610417921968128206848011468802621446811741
51950251971040845760019973120280371202339635210747904209715294875313
61136689130955846547212818553139232593920036352819225139609698566144167772161321442641
76625109856959764891398401616992256341184921647652864004405854208260046848088919244813421772818405321661
838613965554081144355493472013429540864330810941445552298393664306020352507510784002612212531279188459521073741824256353060613
92250586813547702608252731176321073997306883025110425605930213703688256141393928163541647365618521866242563558604806979321856085899345923570537526921
10131173812122532603952176465391936833153784320264118300672059226935787529633549778944114522870251529875824312320602503380992024685324533766098853560326871947673649731172316281
11764537004514214182607212137800656006303246110720222098172723205607260415590410435723316428814496576621772815031138988851211490044149760062941671981056233989818286085291399708672549755813888692665874901013
1244560482149891481312840824334961024046696563404800181043689512960508191469535232106509068245401616932001755955202051718644760576188586873428377612943850314137606433173814640642188715334041604562973255270443980465111049647591076297901
13259717522849556322990816055378168098560339822931200000143762482704384044414257291591681035171861784166418534960746987520257056018834391042761406112228966422790318390771712141915855582658566454476852428800202420090673561639142613948825635184372088832134373609193269601
141513744654945345656383189123685314982464002435266368583680111632887223500803764136152177049696576626857869312192330313249062912300461578631774208369443313066967040356360014291533824266828743094304768152040674047295488637353905272913921853776604430336033425153484390402814749767106561871582937629476513
Technical details
Node id
n-246441fdbcd28bec · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-cf47c94c08cce238

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:        29,         88,          64;
  1:       169,        752,        1088,          512;
  2:       985,       5776,       12480,        11776,         4096;
  3:      5741,      41784,      120000,       169984,       118784,        32768;
  4:     33461,     290856,     1041792,      1968128,      2068480,      1146880,       262144;
  5:    195025,    1971040,     8457600,     19973120,     28037120,     23396352,     10747904,      2097152;
  6:   1136689,   13095584,    65472128,    185531392,    325939200,    363528192,    251396096,     98566144,     16777216;
  7:   6625109,   85695976,   489139840,   1616992256,   3411849216,   4765286400,   4405854208,   2600468480,    889192448,    134217728;
  8:  38613965,  554081144,  3554934720,  13429540864,  33081094144,  55522983936,  64306020352,  50751078400,  26122125312,   7918845952,  1073741824;
  9: 225058681, 3547702608, 25273117632, 107399730688, 302511042560, 593021370368, 825614139392, 816354164736, 561852186624, 256355860480, 69793218560, 8589934592;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 29 + 88*x + 64*x^2, P(1,x) = 169 + 752*x + 1088*x^2 + 512*x^3.
%t Mathematica
P[0, x_] := 29 + 88*x + 64*x^2;
P[1, x_] := 169 + 752*x + 1088*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 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] = 29 + 88*x + 64*x^2; v[2] = 169 + 752*x + 1088*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + 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)

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