Real-rooted atlas

A207606 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-24 + 96\,x - 16\,n + 160\,n x + 8\,n^{2} + 64\,n^{2} x\right)\,P_{n}(x) = \left(-132 + 336\,x + 768\,x^{2} - 8\,n + 448\,n x + 640\,n x^{2} + 12\,n^{2} + 112\,n^{2} x + 128\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-5 + 56\,x + 368\,x^{2} + 256\,x^{3} + 5\,n + 64\,n x + 208\,n x^{2} + 128\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 44 + 64\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
04464108
11765443841104
262029764416204810060
320401334430720296961024086040
464325337616780825088017920049152706848
5196961982407929601617920178176010076162293765647568
6590446989123400512883200013255680115261445390336104857644211204
717416023708481360204843014144821452809719808069730304277872644718592340740720
8507220780691251630720192754688448250880674365440657391616401080320139198464209715202593957780
9146212825107648188131584810613760222638080040775024645049483264418067251222177382406815744009227468819550941488
1041792647921305666340992032424550401028396032022244769792335194030083522428928025327042560118803660803275751424402653184146147492928
111186133624596563222773765121245493760044837775360112019816448199317651456254723751936232272691200147524157440620085903361550214758417448304641084941553304
123346166075357452876445134084626836582418650405376052903600128010871926947841640113373184181774791475214630151782408318563123203166867292167241046425675161927688006778829980
139390872022824512322517761088016712089804874601621504023703076454405531095105536963564011520012601481232384123204009984008870827130880456447649382415877957222403344705781763221225472058789398360720
142623628126845048928815830266245893749196802887149081600101627647672322656752830054452551813169152793470763008009150477107200079990389473280520858065960962447037484236878356309606401530082099200137438953472429748890974428
Technical details
Node id
n-89f0132b57ffc5a7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-96ad4e4ff7c13b82

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:      44,       64;
  1:     176,      544,       384;
  2:     620,     2976,      4416,      2048;
  3:    2040,    13344,     30720,     29696,      10240;
  4:    6432,    53376,    167808,    250880,     179200,      49152;
  5:   19696,   198240,    792960,   1617920,    1781760,    1007616,     229376;
  6:   59044,   698912,   3400512,   8832000,   13255680,   11526144,    5390336,    1048576;
  7:  174160,  2370848,  13602048,  43014144,   82145280,   97198080,   69730304,   27787264,    4718592;
  8:  507220,  7806912,  51630720, 192754688,  448250880,  674365440,  657391616,  401080320,  139198464,  20971520;
  9: 1462128, 25107648, 188131584, 810613760, 2226380800, 4077502464, 5049483264, 4180672512, 2217738240, 681574400, 92274688;
%F Formula
The row polynomials satisfy (8*n^2-16*n-24+(64*n^2+160*n+96)*x)*P(n,x) = (12*n^2-8*n-132+(112*n^2+448*n+336)*x+(128*n^2+640*n+768)*x^2)*P(n-1,x) + (5*n-5+(64*n+56)*x+(208*n+368)*x^2+(128*n+256)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 44 + 64*x.
%t Mathematica
P[0, x_] := 44 + 64*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 8*n - 132 + (112*n^2 + 448*n + 336)*x + (128*n^2 + 640*n + 768)*x^2)*P[n - 1, x] + (5*n - 5 + (64*n + 56)*x + (208*n + 368)*x^2 + (128*n + 256)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 16*n - 24 + (64*n^2 + 160*n + 96)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 44 + 64*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n^2 - 8*n - 132 + (112*n^2 + 448*n + 336)*x + (128*n^2 + 640*n + 768)*x^2)*v[i - 1] + (5*n - 5 + (64*n + 56)*x + (208*n + 368)*x^2 + (128*n + 256)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 16*n - 24 + (64*n^2 + 160*n + 96)*x)); 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