Real-rooted atlas

A207606 → x ↦ x+1 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 26 + 60\,x + 32\,x^{2}\)
\(P_{1}(x) = 97 + 336\,x + 368\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0266032118
197336368128929
23621672278419845127314
31351780017456189449984204857583
450423493298240143616115200481288192453350
5188171520965152329484801025280651264225280327683569217
67022664872025710725711232777984066580483497984103219213107228100386
726208727236881236393632180736529390085710028840398848180879364653056524288221233871
89781221129438057779424172467456332699136433499136386498560232914944908328962070937620971521741770582
93650401463663202639312808888067841967727360300769280032395919362459566080129033830444564480091226112838860813712930785
10136234821887725041183410976443848480011093437440194681815042460264038422563717120149087846406923223040214538649639845888033554432107961675698
1150843527763217624522480264021598776320601599616001192387829761730436956161862058639361485996687368688238592036183212032101669928961728053248134217728849980474799
121897506263067472100227686700801028558310403159415142406981267968001144527273984141443452108813243133460489350049955844901170053121850023608324754663014474490839045368709126691882122694
137081589771226567328098119766080480899228160161502941952039370344611847197572587520100496413163521080639179980889503909806085664304791552269031047168092846791065621984863846431943819264214748364852685076506753
14264288528248827856928418773087360221322014592080677730764802151012872601643393900920832675744210944008209981911859278092126126080579686660833283323345869209614427566899200458581711257610066329600001363652116488589934592414788729931330
Technical details
Node id
n-3d7f7a3a8ead9e51 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-cf183fd17be494dc

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:      26,       60,        32;
  1:      97,      336,       368,       128;
  2:     362,     1672,      2784,      1984,        512;
  3:    1351,     7800,     17456,     18944,       9984,       2048;
  4:    5042,    34932,     98240,    143616,     115200,      48128,       8192;
  5:   18817,   152096,    515232,    948480,    1025280,     651264,     225280,      32768;
  6:   70226,   648720,   2571072,   5711232,    7779840,    6658048,    3497984,    1032192,     131072;
  7:  262087,  2723688,  12363936,  32180736,   52939008,   57100288,   40398848,   18087936,    4653056,    524288;
  8:  978122, 11294380,  57779424, 172467456,  332699136,  433499136,  386498560,  232914944,   90832896,  20709376,  2097152;
  9: 3650401, 46366320, 263931280, 888806784, 1967727360, 3007692800, 3239591936, 2459566080, 1290338304, 445644800, 91226112, 8388608;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 26 + 60*x + 32*x^2, P(1,x) = 97 + 336*x + 368*x^2 + 128*x^3.
%t Mathematica
P[0, x_] := 26 + 60*x + 32*x^2;
P[1, x_] := 97 + 336*x + 368*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 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] = 26 + 60*x + 32*x^2; v[2] = 97 + 336*x + 368*x^2 + 128*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 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