Real-rooted atlas

A207823 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(9 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19413
2807216168
3711964432642171
46319114487728230425628055
55616012734411491251584115201024362544
6499121135928815358569216003097605529640964685017
74435929141027321914494414386240646272017356802580481638460542677
839424240143309016227179568205134336115399680414167049261056117964865536782369784
9350382231143337537225987072322740541056185267174483261337624875827247644672530841626214410110264515
103114015839141585982562889468700834854564096275208102401486279065655600168961422655488238288896235929601048576130651068911
1127675760320138450072288314087868864426529283840385252876800243015743488109242556416349963223047829913600116523008010380902441943041688353631328
122459678270411342595093616335169642192050602604659205145872216320371329040793619496859648007505144709122102162227204178313216055941529604529848321677721621817946138353
132186034683049129287770784203522156030288058522600597120661686389337605376008679321632291092758528145183777751044886173974528121574785024021737609625626449281024196293427267108864281944946167261
1419428344320400123760537344360365357454617680662529386119680824462280576000744802046466048503710496034816259079256539136101838860648448304446434181126813782114304110709492940812344675532884557168642684354563643466354036040
Technical details
Node id
n-44fa69d1dfabf231 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-f243dbc77e21e135

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:         1;
  1:         9,          4;
  2:        80,         72,         16;
  3:       711,        964,        432,         64;
  4:      6319,      11448,       7728,       2304,        256;
  5:     56160,     127344,     114912,      51584,      11520,      1024;
  6:    499121,    1359288,    1535856,     921600,     309760,     55296,      4096;
  7:   4435929,   14102732,   19144944,   14386240,    6462720,   1735680,    258048,    16384;
  8:  39424240,  143309016,  227179568,  205134336,  115399680,  41416704,   9261056,  1179648,   65536;
  9: 350382231, 1433375372, 2598707232, 2740541056, 1852671744, 832613376, 248758272, 47644672, 5308416, 262144;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (9+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 4*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (9 + 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] = 1; v[2] = 9 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (9 + 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