Real-rooted atlas

A093375 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
216815
319242054
4112488048189
511580200240112648
61181204007206722562187
71211687001680235217925767290
812422411203360627271684608128024057
91272881680604814112215042073611520281678732
1013036024001008028224537606912057600281606144255879
11133440330015840517441182721900802112001548806758413312826686
1213652844002376088704236544456192633600619520405504159744286722657205
131396245720343201441444392969884161647360201344017571841038336372736614408503056
1414272872804804822422476876819768323843840563763261501444845568260915286016013107227103491
Technical details
Node id
n-0bb1f53c9b1ccec9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-49d880ced4b36b3a

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: 1,  3;
  2: 1,  6,   8;
  3: 1,  9,  24,   20;
  4: 1, 12,  48,   80,   48;
  5: 1, 15,  80,  200,  240,   112;
  6: 1, 18, 120,  400,  720,   672,   256;
  7: 1, 21, 168,  700, 1680,  2352,  1792,   576;
  8: 1, 24, 224, 1120, 3360,  6272,  7168,  4608,  1280;
  9: 1, 27, 288, 1680, 6048, 14112, 21504, 20736, 11520, 2816;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+(2*n-2)*x^2)*P(n-1,x) - (x^2+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[(1 + 3*x + (2*n - 2)*x^2)*P[n - 1, x] - (x^2 + 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] = (1 + 3*x + (2*n - 2)*x^2)*v[i - 1] - (x^2 + 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.