Real-rooted atlas

A038226 → basis A013609 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-297 + 80\,x\right)\,P_{n-2}(x) + \left(38 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 729 + 1120\,x + 256\,x^{2}\)
\(P_{1}(x) = 19683 + 51632\,x + 28928\,x^{2} + 4096\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
072911202562105
11968351632289284096104339
253144120026241938944638976655365177521
3143489077084270410126080056401920130416641048576256944571
438742048923693412164565737216372878540814296678402537553921677721612751484777
51046035320376341323504187000432384206095134720114626723840332492963844781506560268435456632823206051
628242953648123956352423687161000950784100814317486087527029145600313650891980872900358963288046829568429496729631405380930145
776255974849877371646696923226101637746995245142020123852842977348930764823891490216345679126115385344153126321520641593432866816687194767361558567934514763

All 15 rows

Technical details
Node id
n-2892f947446cb6c0 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-654cf986588e8fef

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:              729,              1120,                256;
  1:            19683,             51632,              28928,               4096;
  2:           531441,           2002624,            1938944,             638976,               65536;
  3:         14348907,          70842704,          101260800,           56401920,            13041664,            1048576;
  4:        387420489,        2369341216,         4565737216,         3728785408,          1429667840,          253755392,           16777216;
  5:      10460353203,       76341323504,       187000432384,       206095134720,        114626723840,        33249296384,         4781506560,          268435456;
  6:     282429536481,     2395635242368,      7161000950784,     10081431748608,       7527029145600,      3136508919808,       729003589632,        88046829568,        4294967296;
  7:    7625597484987,    73716466969232,    261016377469952,    451420201238528,     429773489307648,    238914902163456,     79126115385344,     15312632152064,     1593432866816,      68719476736;
  8:  205891132094649,  2234326000525792,   9162919352372480,  18908924533309440,   22125102697152512,  15625761293598720,   6863837466722304,   1880068246732800,   311320704450560,   28449863368704,   1099511627776;
  9: 5559060566555523, 66914903242431536, 312315604647529728, 751955352333471744, 1051767584799457280, 911204725513256960, 506450740336656384, 183046230325067776, 42663039727566848, 6169291023974400, 502476813893632, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = -(297-80*x)*P(n-2,x) + (38+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 729 + 1120*x + 256*x^2, P(1,x) = 19683 + 51632*x + 28928*x^2 + 4096*x^3.
%t Mathematica
P[0, x_] := 729 + 1120*x + 256*x^2;
P[1, x_] := 19683 + 51632*x + 28928*x^2 + 4096*x^3;
P[n_, x_] := P[n, x] = Expand[-(297 - 80*x)*P[n - 2, x] + (38 + 16*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] = 729 + 1120*x + 256*x^2; v[2] = 19683 + 51632*x + 28928*x^2 + 4096*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(297 - 80*x)*v[i - 2] + (38 + 16*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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.