Real-rooted atlas

A123519 → basis A007318 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 29 + 44\,x + 16\,x^{2}\)
\(P_{1}(x) = 169 + 376\,x + 272\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
029441689
116937627264881
29852888312014722568721
357412089230000212487424102486329
433461145428260448246016129280358404096854569
5195025985520211440024966401752320731136167936163848459361
6113668965477921636803223191424203712001136025639280647700486553683739041
766251094284798812228496020212403221324057614891520068841472203161603473408262144828931049
83861396527704057288873368016786926082067568384173509324810047815683964928001020395521546649610485768205571449
922505868117738513046318279408134249663361890694016018531917824129002209286377766912219473510450069504068157440419430481226783441
1013117381211126630197644116347984104144223040165073937920185084174336150524215296894709923843857743872011767644160241067622429779558416777216804062262961
11764537004571070913036303445016400787905763840138811357952017522688798721630581768192113254504857658715386675222441492480061466476544114252840961291845632671088647959395846169
1244560482149445740656420206083740256058370704256001131523059456015880983422976166420419133441322812637184080145259560963683337371648126404788224031411981516853435432960557003571226843545678789896198729
132597175228492781614954080138445420246404247786640000089851551690240138794554036224161745603403776144804380835840100412507357184539337131294722225617030348869294851358721575800012800247094837248238907555841073741824779939566141121
14151374465494517282819159456921328745616003044082960729606977055451468801176292547555328150900979465420815025805722583041173678041530368721568970833920348007826456576130287472214016371193051873287780199038976113145544704010200547328042949672967720605765212481
Technical details
Node id
n-91aecd3506520caf · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-9247102c6b6e4eea

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:        29,         44,         16;
  1:       169,        376,        272,          64;
  2:       985,       2888,       3120,        1472,         256;
  3:      5741,      20892,      30000,       21248,        7424,        1024;
  4:     33461,     145428,     260448,      246016,      129280,       35840,        4096;
  5:    195025,     985520,    2114400,     2496640,     1752320,      731136,      167936,      16384;
  6:   1136689,    6547792,   16368032,    23191424,    20371200,    11360256,     3928064,     770048,      65536;
  7:   6625109,   42847988,  122284960,   202124032,   213240576,   148915200,    68841472,   20316160,    3473408,    262144;
  8:  38613965,  277040572,  888733680,  1678692608,  2067568384,  1735093248,  1004781568,  396492800,  102039552,  15466496,  1048576;
  9: 225058681, 1773851304, 6318279408, 13424966336, 18906940160, 18531917824, 12900220928, 6377766912, 2194735104, 500695040, 68157440, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 29 + 44*x + 16*x^2, P(1,x) = 169 + 376*x + 272*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 29 + 44*x + 16*x^2;
P[1, x_] := 169 + 376*x + 272*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 4*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] = 29 + 44*x + 16*x^2; v[2] = 169 + 376*x + 272*x^2 + 64*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + 4*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

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