Real-rooted atlas

A078812 → basis A013609 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(5 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
25813
3248064168
41155849605122171
55513760940810240409628055
62640226247616012595210240032768362544
712649130480552384122880015155209830402621444685017
86060573096837296001043712017305600170065929175040209715260542677
9290376400920023943360807936001685094402224947201816657928388608016777216782369784
10139127521638040148060800585077760147159040024435425282679111680187065958475497472013421772810110264515
1166659991153112008894649604029081600118700646402376794112032762232832307023052801872337305667108864001073741824130651068911
1231938720608245952522175360026676049920901113856002113566801923512755814404137387294723384803328001832071987205905580032085899345921688353631328
131530276013181428320300852706561711251968006520952627201753906544640341446911590448481959936004983588126720361716776960017598628495365153960755206871947673621817946138353
147331992851652311645617065602624010696320993284539366502400137749381447683075232235520051142994165760633650282496005777133666304037677600604160166472932392964466765987840549755813888281944946167261
153512968824852997482409553797921926542283509760306017940439041034357161984002605466478182404968853536768007209855064473607921597415424006487988334755843841418749542401554434563768323848290697216043980465111043643466354036040
Technical details
Node id
n-2ba30707749e6984 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-852d548e80d9ee9b

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:       5,        8;
   3:      24,       80,        64;
   4:     115,      584,       960,       512;
   5:     551,     3760,      9408,     10240,       4096;
   6:    2640,    22624,     76160,    125952,     102400,      32768;
   7:   12649,   130480,    552384,   1228800,    1515520,     983040,     262144;
   8:   60605,   730968,   3729600,  10437120,   17305600,   17006592,    9175040,    2097152;
   9:  290376,  4009200,  23943360,  80793600,  168509440,  222494720,  181665792,   83886080,  16777216;
  10: 1391275, 21638040, 148060800, 585077760, 1471590400, 2443542528, 2679111680, 1870659584, 754974720, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+8*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 8*x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 8*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] = 5 + 8*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (5 + 8*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

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