Real-rooted atlas

A086645 → basis A013611 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 128\,x - 256\,x^{2}\right)\,P_{n-2}(x) + \left(12 + 32\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
261622
356256256568
457638407680409616192
560165324816384019660865536485248
662976696320290816055705604587520104857614874112
765945687490564620288012215910416515072010066329616777216460361728
86905856106889216682622976230477004843599790084462739456211392921626843545614306271232
972318976127926272095839846403939290316895965675520139049566208112742891520429496729604294967296445331243008
107573340161507118284812957253632062778664550418726392954883519994134528408129267302427058293964808504035246086871947673613872066199552
11793090457617536385024017015452467209495233167360335296017203207757140433305611744588070912011235634446336062362925137920164926744166401099511627776432238415577088
1283053510656202007746969621834036674560137904152117248562713193349120154707298982297628963799155343363639727085322240293899458104524813908822091366403144603255439361759218604441613469663805571072
13869747654656230776645877762749493107752961938924331073536897803213183385628607986884673536638710427743682569967182856952217610629363734780313673728851712147456301881912522178565910974510923776281474976710656419769841214291968
14910811568537626181181112320034087007944704002654710695723008013753730822307840049850895130689536012934860947587072002414912363665817600321063992380358656029559270601129984001786134649089556480640355572016742400109775240917155840450359962737049613082007491200417792
15953814257500162952631102210048417066020503552003555195256989286402037737269263073280828002074703691776024491984347547566080532766160215605248008499519551824723968097910366961359912960788918064724626636804200732552429830144013319395897948241920201761263306198220872057594037927936407700393127101595648
Technical details
Node id
n-3076284606182517 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-261d320b22506b21

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         1;
   2:         6,          16;
   3:        56,         256,          256;
   4:       576,        3840,         7680,         4096;
   5:      6016,       53248,       163840,       196608,         65536;
   6:     62976,      696320,      2908160,      5570560,       4587520,       1048576;
   7:    659456,     8749056,     46202880,    122159104,     165150720,     100663296,      16777216;
   8:   6905856,   106889216,    682622976,   2304770048,    4359979008,    4462739456,    2113929216,     268435456;
   9:  72318976,  1279262720,   9583984640,  39392903168,   95965675520,  139049566208,  112742891520,   42949672960,   4294967296;
  10: 757334016, 15071182848, 129572536320, 627786645504, 1872639295488, 3519994134528, 4081292673024, 2705829396480, 850403524608, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = -(16+128*x+256*x^2)*P(n-2,x) + (12+32*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 16*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(16 + 128*x + 256*x^2)*P[n - 2, x] + (12 + 32*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 16*x; for(n = 3, N, my(i = n); v[i] = -(16 + 128*x + 256*x^2)*v[i - 2] + (12 + 32*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, v[i])))};
T(10)

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