Real-rooted atlas

image under θ+1 → (θ+1)² → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n^{2} - 2\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-n^{2} - 6\,n^{2} x - 9\,n^{2} x^{2} - 2\,n^{3} - 12\,n^{3} x - 18\,n^{3} x^{2} - n^{4} - 6\,n^{4} x - 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2 + 6\,x - 4\,n^{2} - 12\,n^{2} x + 2\,n^{4} + 6\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
182432
227162243432
364576172817284096
412515006750135001012532000
5216324019440583208748052488221184
63436174463051852204167455000942500471404928
7512107529676848384014515202612736261273611197448388608

All 15 rows

Technical details
Node id
n-fc48bc04eb0177e8 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f9762973ec1089d7

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:    8,    24;
  2:   27,   162,    243;
  3:   64,   576,   1728,    1728;
  4:  125,  1500,   6750,   13500,    10125;
  5:  216,  3240,  19440,   58320,    87480,    52488;
  6:  343,  6174,  46305,  185220,   416745,   500094,   250047;
  7:  512, 10752,  96768,  483840,  1451520,  2612736,  2612736,  1119744;
  8:  729, 17496, 183708, 1102248,  4133430,  9920232, 14880348, 12754584,  4782969;
  9: 1000, 27000, 324000, 2268000, 10206000, 30618000, 61236000, 78732000, 59049000, 19683000;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n^2)*P(n,x) = -(n^4+2*n^3+n^2+(6*n^4+12*n^3+6*n^2)*x+(9*n^4+18*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^2+2+(6*n^4-12*n^2+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 + n^2 + (6*n^4 + 12*n^3 + 6*n^2)*x + (9*n^4 + 18*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^2 + 2 + (6*n^4 - 12*n^2 + 6)*x)*P[n - 1, x])/(n^4 - 2*n^3 + n^2)]];
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] = 8 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 + n^2 + (6*n^4 + 12*n^3 + 6*n^2)*x + (9*n^4 + 18*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^2 + 2 + (6*n^4 - 12*n^2 + 6)*x)*v[i - 1])/(n^4 - 2*n^3 + n^2)); 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. [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.