Real-rooted atlas

A126351 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
24131027
38437137159
4161293463831511025
5323711449256621296747221
66410495636142441869112360326355307
7128295521111716111323341372437524217007457631
825683537763834032583167112035931028392480513948284065569
951223779283445156551648645239218539109184177912582321611356259538566021
10102468265103405670704582720302364865723998988779984958462706535225245113535027388757083
1120481976273784659316357371478752324317698788305700471073812026926416930512579684164776875234308404146851583
1240965764971393034614106377978991798727693032506472973508103959691541155419122187544638604323868550125676405216325488546636281185
1381921692627516367536293256984177013125173256307524823374741893597520160128683916932125198595237844598324563763709514699757639661192059223551163837685
141638449959611928995002816496448219767716751066321320343482768612198002401302601323593018034158770292823913716871667088330861805103379263399088226739357790971836026825500514059
Technical details
Node id
n-e0bf38819e6d5bc8 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-91f3a2ca42629500

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,    13,     10;
  3:   8,    43,     71,      37;
  4:  16,   129,    346,     383,     151;
  5:  32,   371,   1449,    2566,    2129,     674;
  6:  64,  1049,   5636,   14244,   18691,   12360,     3263;
  7: 128,  2955,  21111,   71611,  132334,  137243,    75242,   17007;
  8: 256,  8353,  77638,  340325,  831671, 1203593,  1028392,  480513,   94828;
  9: 512, 23779, 283445, 1565516, 4864523, 9218539, 10918417, 7912582, 3216113, 562595;
%F Formula
The row polynomials satisfy P(n,x) = (2-(n-4)*x-(n-1)*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx 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 - (n - 4)*x - (n - 1)*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], 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 - (n - 4)*x - (n - 1)*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], x)); 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

Candidate equalities ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.