Real-rooted atlas

A013609 → basis A008288 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(2 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
2416929
38528227169
41614439235281985
5323681440228013782435741
66489645921067211404510472933461
7128211213408414886573651740182262187195025
82564864368641431043049923572162194086342465611136689
95121100897024454144122515219414081773952885712216514196836625109
1010242457624704013555204442496896409611112992823440034445007282725904938613965
1120485427261286438602241492428836786048582897285864444836276168130104362421010177147225058681
124096118784148889610592256472593921380876802683243523457815042902599521533113444800680879716165314411311738121
13819225804835553282820300814276966448348083211177986561771081728190606563213642814886264513601737811762604061015943237645370045
14163845570568368128732487684150691841600782336430391782481357437441078304812898960008966145871504248944737661912873284499120478296944560482149
Technical details
Node id
n-0e4e8cd316890790 · revision 0 · JSON
Definition source
derived:reversal
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:   2,     3;
  2:   4,    16,     9;
  3:   8,    52,    82,     27;
  4:  16,   144,   392,    352,      81;
  5:  32,   368,  1440,   2280,    1378,     243;
  6:  64,   896,  4592,  10672,   11404,    5104,     729;
  7: 128,  2112, 13408,  41488,   65736,   51740,   18226,   2187;
  8: 256,  4864, 36864, 143104,  304992,  357216,  219408,  63424,   6561;
  9: 512, 11008, 97024, 454144, 1225152, 1941408, 1773952, 885712, 216514, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (2*x-3*x^2)*P(n-2,x) + (2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2*x - 3*x^2)*P[n - 2, x] + (2 + 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] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2*x - 3*x^2)*v[i - 2] + (2 + 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. [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.