Real-rooted atlas

A123519 → basis A007318 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 29 + 66\,x + 36\,x^{2}\)
\(P_{1}(x) = 169 + 564\,x + 612\,x^{2} + 216\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0296636131
11695646122161561
2985433270204968129618601
35741313386750071712375847776221651
433461218142586008830304654480272160466562641211
519502514782804757400842616088711205552064191289627993631472881
61136689982168836828072782710561031292008626694444743104131569921679616375033361
76625109642719822751411606821686081079530416113082480078414739234712064089019648100776964468927451
83861396541556085819996507805665587552104670649441317586435211445090048677445120026151621125945840646046617653252096051
9225058681266077695614216128668453092613849571638456014072675097614694157900810897012684856248660224192483993603930301440362797056634556225161
10131173812116899452964992617829643514867527608356868107201405482948864171456488985615286957839369886975603204523877734401390117386242575859097621767823367561422605881
1176453700451066063695546827512869002659181952960702732499632013306291806528185733454533121935059391590415048111405312862726360320035444667709449882591805441676122398721306069401690102515045411
124456048214966861098463046368841557601970011268640057283354884960120596217868224189563258669184226014940431360205403534366976141599862277632728913705062402717059711795269330517401601084037603328783641640961073668757939051
1325971752284941724224311203115021955544014336279910000045487348043184010539711447125761842383513771136247411860068736025734627373847042073393116264448128340292993228859938354656460820445428090880048089475366912697441060454447018498457612793922580223201
1415137446549452592422873918420729896776360010273779992462403532134322306080893247153299827217188564692233088256729977463196163008008449406540827739535259617280200678849066741761126954826186342448160939131002881514178500050944330304951664640446675735347202821109907456152453402204739361
Technical details
Node id
n-15fb5012236a46df · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-3bef121b79a49566

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,         66,          36;
  1:       169,        564,         612,         216;
  2:       985,       4332,        7020,        4968,        1296;
  3:      5741,      31338,       67500,       71712,       37584,         7776;
  4:     33461,     218142,      586008,      830304,      654480,       272160,        46656;
  5:    195025,    1478280,     4757400,     8426160,     8871120,      5552064,      1912896,       279936;
  6:   1136689,    9821688,    36828072,    78271056,   103129200,     86266944,     44743104,     13156992,     1679616;
  7:   6625109,   64271982,   275141160,   682168608,  1079530416,   1130824800,    784147392,    347120640,    89019648,    10077696;
  8:  38613965,  415560858,  1999650780,  5665587552, 10467064944,  13175864352,  11445090048,   6774451200,  2615162112,   594584064,   60466176;
  9: 225058681, 2660776956, 14216128668, 45309261384, 95716384560, 140726750976, 146941579008, 108970126848, 56248660224, 19248399360, 3930301440, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 29 + 66*x + 36*x^2, P(1,x) = 169 + 564*x + 612*x^2 + 216*x^3.
%t Mathematica
P[0, x_] := 29 + 66*x + 36*x^2;
P[1, x_] := 169 + 564*x + 612*x^2 + 216*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 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] = 29 + 66*x + 36*x^2; v[2] = 169 + 564*x + 612*x^2 + 216*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + 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

Same fingerprint ?

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