Real-rooted atlas

A013609 → basis A013610 → basis A008288

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 18\,x\right)\,P_{n-2}(x) + \left(12 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19615
28114436261
3729221414042164563
46561285123175211664129679785
5590493348545540403499208942477761395063
653144137208168313516777988833475686531844665624393069
747829693986026211315538014376463290734256293232964618944279936426519675
8430467214161248641439535888234369417620010525129389053442418647043191270416796167457816529
93874204894261645062174334692963491669721638212733664241146668168938916352190972339221667046410077696130402020447
1034867844014300373332820339428050048623308560065621106460852932090470426232494630480042100480145874649601451188224604661762280116019285
113138105960942887465847023047597060926426214332696103886880298561032585044601665165380450562643792846336683827100928108567019008961412198436279705639868470163107
122824295364814236443578656255127860503288147593396670415425603915083218376246826726414259856195123273385454746880251155785438725629159120896791260379136631266877442176782336697111418762361
1325418658283294151714345703027706332586384899876581162928021751053547926243038884276607136282367633383014417821321066298887708246105167362276209590174724496470432358456683412029444114118615041306069401612189189306254535
142287679245496140415667148728029612735321653081190694962653934429395512881615376473322655095402245157498492173203238913010119495936205855055307394567656492288835584198526303992729635034658663219240024496812032266438157926478364164096213131404743720381
Technical details
Node id
n-b3709c7e6da5a05e · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

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 0)
  • 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:         1;
  1:         9,          6;
  2:        81,        144,          36;
  3:       729,       2214,        1404,         216;
  4:      6561,      28512,       31752,       11664,        1296;
  5:     59049,     334854,      554040,      349920,       89424,        7776;
  6:    531441,    3720816,     8313516,     7779888,     3347568,      653184,      46656;
  7:   4782969,   39860262,   113155380,   143764632,    90734256,    29323296,    4618944,     279936;
  8:  43046721,  416124864,  1439535888,  2343694176,  2001052512,   938905344,  241864704,   31912704,   1679616;
  9: 387420489, 4261645062, 17433469296, 34916697216, 38212733664, 24114666816, 8938916352, 1909723392, 216670464, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = -(27-18*x)*P(n-2,x) + (12+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(27 - 18*x)*P[n - 2, x] + (12 + 6*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] = 1; v[2] = 9 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(27 - 18*x)*v[i - 2] + (12 + 6*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

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.