Real-rooted atlas

A123519 → basis A013610 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(10 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 89 + 228\,x + 144\,x^{2}\)
\(P_{1}(x) = 881 + 3336\,x + 4176\,x^{2} + 1728\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
089228144461
188133364176172810121
2872143704816486739220736222201
3863295383561336752165196810160642488324878301
48545696375804197461443249331229963520146810882985984107100421
5845936173474512272634336560234880688538880506124288206032896358318082351330961
6837390418298816483588291360884146752013578243840133090283528130834432283071283242998169651622180721
78289310499230210460455688590401309139366402411915097602955230853122408106516481258413096963826837094451597803521133336644901
882055714491014193955405628628245601847124207360396930409344058362199418885946252705792414531020390418923494440965108182548486191736422424881784007101
98122678344111034306023286800092133040250946820316806161733991296010569832545484812925635570892811268229319884868628948516864278112160972806748992700416743008370688546265911511321
1080406226296111907608029032806792257337763307008017059209133402794163201790555113562112255499720984166426737538902917122036584654110720110114872492032040116260280729688417996111872891610044825611992968269241961
11795939584616912762139684352494288346155310442500640438328321304019507472128028759936163162112469073771054530565728482312781824052382264276090880354226918824345601721866173441638456973881864355841150176957825024106993205379072263299036011811821
12787898961987291359819110560236108796121519195525348454117525964818048937899379046444229114741471436881163800780263424011330630026533273601209203935640616960981712941412515840596857757330571264263509404681240576798615117150289921487205554769100812839184645488645780585823990618101
137799395661411211441604846314358412417106738436524866115069353179827224336680889662991366560024085909467136133769664698974617602101300129703706624025636413123970007040242921539499301273601773191420852148633697916897465928253443959577796367351808110695170285187891219130385121778073615407021574586368126909589091781786401
147720605765212481152159940314569056140382364324945593680480752027551060483208999983305078016094361966779275546624211678315722085588992369520547606486876160507310942868504248320549576784045528842240468228131727045820416310436358563504848896157016193411072393216585695785293793198081519517502793580544024497164303592325121848842588950364162786230374195208682721
Technical details
Node id
n-34c9cab803cd1810 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-63234ffc8ab36e77

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:          89,           228,           144;
  1:         881,          3336,          4176,           1728;
  2:        8721,         43704,         81648,          67392,          20736;
  3:       86329,        538356,       1336752,        1651968,        1016064,          248832;
  4:      854569,       6375804,      19746144,       32493312,       29963520,        14681088,         2985984;
  5:     8459361,      73474512,     272634336,      560234880,      688538880,       506124288,       206032896,        35831808;
  6:    83739041,     829881648,    3588291360,     8841467520,    13578243840,     13309028352,      8130834432,      2830712832,      429981696;
  7:   828931049,    9230210460,   45568859040,   130913936640,   241191509760,    295523085312,    240810651648,    125841309696,    38268370944,     5159780352;
  8:  8205571449,  101419395540,  562862824560,  1847124207360,  3969304093440,   5836219941888,   5946252705792,   4145310203904,  1892349444096,   510818254848,   61917364224;
  9: 81226783441, 1103430602328, 6800092133040, 25094682031680, 61617339912960, 105698325454848, 129256355708928, 112682293198848, 68628948516864, 27811216097280, 6748992700416, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (10+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 89 + 228*x + 144*x^2, P(1,x) = 881 + 3336*x + 4176*x^2 + 1728*x^3.
%t Mathematica
P[0, x_] := 89 + 228*x + 144*x^2;
P[1, x_] := 881 + 3336*x + 4176*x^2 + 1728*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (10 + 12*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] = 89 + 228*x + 144*x^2; v[2] = 881 + 3336*x + 4176*x^2 + 1728*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (10 + 12*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