Real-rooted atlas

A123519 → basis A013611 → basis A007318

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(12 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 131 + 184\,x + 64\,x^{2}\)
\(P_{1}(x) = 1561 + 3248\,x + 2240\,x^{2} + 512\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
013118464379
11561324822405127561
2186015128052800240644096150841
32216517609201041600710656241664327683009259
426412111085296818533760168366088581120232652826214460034339
531472881150604384308187264349598720237424640965345282175795220971521197677521
63750333612048182688488454848066438461445637304320305548492810331095041992294401677721623893516081
7446892745127427854760746918560001184529428481205609963528166771916836819435520106535321601795162112134217728476672644099
853252096051362837494040111084056160020123263160322388718194688194144511590410941418700804221986406401067534254081597190963210737418249509559365899
96345562251614752638842128161580948355203291618734233644642667868160423254192291842862444393267213808865116160465683506790410455561011201406601789448589934592189714514673881
1075614226058816174527841278423080740820166452224668047616079665279496192086310492857958468100253915545639427973421465616624618831872049785381847040100512972144641228360646656687194767363784780734111721
1190102515045411796682082958328324749303088672080805032439848961369316431548416016688156083421184150482454545694721016556825870336051445352984739841926348532613120518757961170944951421155409921065151889408054975581388875505900167560539
1210736687579390511021925983745044045112565626105600122423736494436352228165344942727168308940082596347904313403191583047680241978503006781440142892723462995968642226793973022722162583249774182452905407152128008888864315801609180922091929643980465111041506333222617099059
131279392258022320113042378602995936061985737318198400018219048596980961283703680866952732672551591559461457100862173107143127859205400802000487055360364539617031172915219118875919389818887727726671901491202363980064489472005298031138177024082122523478589447872503254876163518437208883230051158552174420641
141524534022047393611657217553163847488843656620079737728026699293565338589184587912439360748339209551149398840036556811842165013708695142411430613121734017024086808277324174196736520415977863643136002454674690929590272090136668741933465602526058901741240320522297710007549952751406246422118406720215068966912281474976710656599516837820871313761
Technical details
Node id
n-4408949914450811 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-0723ba409d749fff

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          131,           184,             64;
  1:         1561,          3248,           2240,            512;
  2:        18601,         51280,          52800,          24064,           4096;
  3:       221651,        760920,        1041600,         710656,         241664,          32768;
  4:      2641211,      10852968,       18533760,       16836608,        8581120,        2326528,         262144;
  5:     31472881,     150604384,      308187264,      349598720,      237424640,       96534528,       21757952,        2097152;
  6:    375033361,    2048182688,     4884548480,     6643846144,     5637304320,     3055484928,     1033109504,      199229440,      16777216;
  7:   4468927451,   27427854760,    74691856000,   118452942848,   120560996352,    81667719168,    36819435520,    10653532160,    1795162112,     134217728;
  8:  53252096051,  362837494040,  1110840561600,  2012326316032,  2388718194688,  1941445115904,  1094141870080,   422198640640,  106753425408,   15971909632,   1073741824;
  9: 634556225161, 4752638842128, 16158094835520, 32916187342336, 44642667868160, 42325419229184, 28624443932672, 13808865116160, 4656835067904, 1045556101120, 140660178944, 8589934592;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (12+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 131 + 184*x + 64*x^2, P(1,x) = 1561 + 3248*x + 2240*x^2 + 512*x^3.
%t Mathematica
P[0, x_] := 131 + 184*x + 64*x^2;
P[1, x_] := 1561 + 3248*x + 2240*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (12 + 8*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] = 131 + 184*x + 64*x^2; v[2] = 1561 + 3248*x + 2240*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (12 + 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 - 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