Real-rooted atlas

A038279 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12911
243681121
381084867291331
41628819445832656114641
5327206480291606561059049161051
6641728194401166403936607085885314411771561
7128403254432408240183708049601167440174478296919487171
825692161451521306368734832026453952595213927652750443046721214358881
9512207363732483919104264539521190427843571283526887475367748409783874204892357947691
1010244608093312011197440881798404761711361785641760459165024077484097807748409780348678440125937424601
112048101376228096030792960277136640174596083278568237442525407632056821671720852325075807670925682231381059609285311670611
1240962211845474304821145608314099205986151424314272949761212195663363409300303206818600606409205110818647531454306162824295364813138428376721
1381924792321293926421349785624018508801945499212811672995276852528478745617728361576644432090394160797776270948897908905980087343167948506254186582832934522712143931
1416384103219230191616543449088672518246460526642176408554834688210113914982482732354024322481970620729655844338966416913816455814081028043512790847117224319321222876792454961379749833583241
Technical details
Node id
n-24d208fb12b74cc6 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-0205823d5b324a40

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,     9;
  2:   4,    36,     81;
  3:   8,   108,    486,     729;
  4:  16,   288,   1944,    5832,     6561;
  5:  32,   720,   6480,   29160,    65610,     59049;
  6:  64,  1728,  19440,  116640,   393660,    708588,    531441;
  7: 128,  4032,  54432,  408240,  1837080,   4960116,   7440174,   4782969;
  8: 256,  9216, 145152, 1306368,  7348320,  26453952,  59521392,  76527504,  43046721;
  9: 512, 20736, 373248, 3919104, 26453952, 119042784, 357128352, 688747536, 774840978, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (2+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + 9*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; for(n = 1, N, my(i = n + 1); v[i] = (2 + 9*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. [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.