Real-rooted atlas

A116414 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(248\,n + 64\,n x - 128\,n x^{2} + 128\,n^{3}\right)\,P_{n}(x) = \left(732 + 272\,x - 448\,x^{2} - 256\,x^{3} + 1372\,n + 784\,n x - 448\,n x^{2} - 256\,n x^{3} + 960\,n^{2} + 768\,n^{2} x + 320\,n^{3} + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(91 - 32\,x - 1056\,x^{2} - 1024\,x^{3} - 256\,x^{4} + 208\,n + 640\,n x + 256\,n x^{2} + 208\,n^{2} + 640\,n^{2} x + 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19413
2627216150
3371756432641623
4204961526096230425616857
510768432486281640832115201024170208
6547332768885372645094402457605529640961683477
7271653166226840648805136960360064013793282580481638416390161
813244069523560282080804488704041832960232099847368704117964865536157599918
96367895526654201836187203539078404147852803053352961400340483794534453084162621441500230403
10302793572832864801137932640258291072036645772803387912192205767475280373350418992332823592960104857614162871789
1114267612814902879686783158720177514022402965637376033164285952252647915521304779161644364595209293004801038090244194304132774531264
1266721840976983135843918070915211627927168022401055872029486492467227148731187217565548544078856519680237318963204463788032452984832167772161237365759609
13310029377739166221172220505629344732386498688160059965312024292026685442631290388480205590626304011550700339204584033484801237172879362111412633619629342726710886411472492455973
141432693671819671139226412141963202724464942896128109254426577921881131088896023487020175360215772895641601462791340032072582981222402580450115584631192420352985829212168455716864268435456105896401963686
Technical details
Node id
n-bb4062f74a4de684 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b3de6817e760bec2

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:      62,       72,        16;
  3:     371,      756,       432,        64;
  4:    2049,     6152,      6096,      2304,       256;
  5:   10768,    43248,     62816,     40832,     11520,      1024;
  6:   54733,   276888,    537264,    509440,    245760,     55296,      4096;
  7:  271653,  1662268,   4064880,   5136960,   3600640,   1379328,    258048,    16384;
  8: 1324406,  9523560,  28208080,  44887040,  41832960,  23209984,   7368704,  1179648,   65536;
  9: 6367895, 52665420, 183618720, 353907840, 414785280, 305335296, 140034048, 37945344, 5308416, 262144;
%F Formula
The row polynomials satisfy (128*n^3+248*n+64*n*x-128*n*x^2)*P(n,x) = (320*n^3+960*n^2+1372*n+732+(256*n^3+768*n^2+784*n+272)*x-(448*n+448)*x^2-(256*n+256)*x^3)*P(n-1,x) + (208*n^2+208*n+91+(640*n^2+640*n-32)*x+(256*n^2+256*n-1056)*x^2-1024*x^3-256*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((320*n^3 + 960*n^2 + 1372*n + 732 + (256*n^3 + 768*n^2 + 784*n + 272)*x - (448*n + 448)*x^2 - (256*n + 256)*x^3)*P[n - 1, x] + (208*n^2 + 208*n + 91 + (640*n^2 + 640*n - 32)*x + (256*n^2 + 256*n - 1056)*x^2 - 1024*x^3 - 256*x^4)*D[P[n - 1, x], x])/(128*n^3 + 248*n + 64*n*x - 128*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((320*n^3 + 960*n^2 + 1372*n + 732 + (256*n^3 + 768*n^2 + 784*n + 272)*x - (448*n + 448)*x^2 - (256*n + 256)*x^3)*v[i - 1] + (208*n^2 + 208*n + 91 + (640*n^2 + 640*n - 32)*x + (256*n^2 + 256*n - 1056)*x^2 - 1024*x^3 - 256*x^4)*deriv(v[i - 1], x))/(128*n^3 + 248*n + 64*n*x - 128*n*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023