Real-rooted atlas

A123519 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 89 + 114\,x + 36\,x^{2}\)
\(P_{1}(x) = 881 + 1668\,x + 1044\,x^{2} + 216\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
08911436239
1881166810442163809
2872121852204128424129660705
386329269178334188206496635047776967471
485456931879024936536406166418727204587844665615418831
5845936136737256681585847002936043033680158163843219264279936245733825
68373904141494082489707284011051834408486402404159071361270442882211494416796163916322369
78289310494615105230113922147601636424208015074469360923509641637626664329831352321494858241007769662415424079
88205571449507096977701407157061402308905259202480815058401823818731849291019852832385235968739199001699769190460466176994730462895
9812267834415517153011641700023033260313683525396038510837445603303072670464201963055795288033041561626808183014454318781440659081318436279705615853271982241
10804062262961595380401451620169806433444413376002132405708376746352055954847298816399218314037762088870226790479554088051202150681103360391760354304431728496642176782336252657621252961
1179593958461696381069842176223572086538827653125800547910481501219217008089874800509881673292776727270444753768068608020461821982848069184945082880168150993500162781927825408280804921344130606940164026668668065135
1278789896198729679909555280118271990303797988866855676469074561128058618711190413821598356709824126818438719161608852054708229120472345287359616019174080886963205828689036431361286667015045121949743938355218154364682247836416409664174041067789199
137799395661411217208024231571792310427668460913128264383669147478415210425556039369620500075268467084820901510109214784016416407263310208010014223876550784047445613183457280173163224692592644781098509078528966693797941248135125940289536116762604503044701849845761022757988416562049
14772060576521248176079970157284528350955910812363984100600940034438825620056249895656737602948811461852360832330747368315758732828868792781756787201981683370580094720107339215633892352045725403488969318415158025320483635238334031594500096714960675407462492743988207616074759412547584282110990745616299953773597203585
Technical details
Node id
n-25c619a616edcf01 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-80bc8b2a60a22464

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,          114,            36;
  1:         881,         1668,          1044,           216;
  2:        8721,        21852,         20412,          8424,          1296;
  3:       86329,       269178,        334188,        206496,         63504,          7776;
  4:      854569,      3187902,       4936536,       4061664,       1872720,        458784,         46656;
  5:     8459361,     36737256,      68158584,      70029360,      43033680,      15816384,       3219264,       279936;
  6:    83739041,    414940824,     897072840,    1105183440,     848640240,     415907136,     127044288,     22114944,      1679616;
  7:   828931049,   4615105230,   11392214760,   16364242080,   15074469360,    9235096416,    3762666432,    983135232,    149485824,    10077696;
  8:  8205571449,  50709697770,  140715706140,  230890525920,  248081505840,  182381873184,   92910198528,  32385235968,   7391990016,   997691904,   60466176;
  9: 81226783441, 551715301164, 1700023033260, 3136835253960, 3851083744560, 3303072670464, 2019630557952, 880330415616, 268081830144, 54318781440, 6590813184, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (10+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 89 + 114*x + 36*x^2, P(1,x) = 881 + 1668*x + 1044*x^2 + 216*x^3.
%t Mathematica
P[0, x_] := 89 + 114*x + 36*x^2;
P[1, x_] := 881 + 1668*x + 1044*x^2 + 216*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (10 + 6*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 + 114*x + 36*x^2; v[2] = 881 + 1668*x + 1044*x^2 + 216*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (10 + 6*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