Real-rooted atlas

A127773 → θ+1 → x ↦ x+1

palindromicγ-positivenonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n - 3\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(-n - 3\,n x - 3\,n x^{2} - n x^{3} + n^{3} + 3\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-3}(x) + \left(6\,n + 12\,n x + 6\,n x^{2} + 3\,n^{2} + 6\,n^{2} x + 3\,n^{2} x^{2} - 3\,n^{3} - 6\,n^{3} x - 3\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x - 3\,n - 3\,n x - 6\,n^{2} - 6\,n^{2} x + 3\,n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 6\,x\)
\(P_{2}(x) = 18 + 36\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16612
218361872
34012012040320
475300450300751200
5126630126012606301264032
61961176294039202940117619612544
72882016604810080100806048201628836864
8405324011340226802835022680113403240405103680
955049501980046200693006930046200198004950550281600
107267260326708712015246018295215246087120326707260726743424
11936102965148015444030888043243243243230888015444051480102969361916928
12118314196780782602605855859369361093092936936585585260260780781419611834845568
1314701911011466042042010510501891890252252025225201891890105105042042011466019110147012042240
14180025200163800655200180180036036005405400617760054054003603600180180065520016380025200180029491200
Technical details
Node id
n-95739f9b92fbd6aa · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:   6,    6;
  2:  18,   36,    18;
  3:  40,  120,   120,    40;
  4:  75,  300,   450,   300,    75;
  5: 126,  630,  1260,  1260,   630,   126;
  6: 196, 1176,  2940,  3920,  2940,  1176,   196;
  7: 288, 2016,  6048, 10080, 10080,  6048,  2016,   288;
  8: 405, 3240, 11340, 22680, 28350, 22680, 11340,  3240,  405;
  9: 550, 4950, 19800, 46200, 69300, 69300, 46200, 19800, 4950, 550;
%F Formula
The row polynomials satisfy (n^3-3*n^2+2*n)*P(n,x) = (n^3-n+(3*n^3-3*n)*x+(3*n^3-3*n)*x^2+(n^3-n)*x^3)*P(n-3,x) - (3*n^3-3*n^2-6*n+(6*n^3-6*n^2-12*n)*x+(3*n^3-3*n^2-6*n)*x^2)*P(n-2,x) + (3*n^3-6*n^2-3*n+6+(3*n^3-6*n^2-3*n+6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 6*x, P(2,x) = 18 + 36*x + 18*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[2, x_] := 18 + 36*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n + (3*n^3 - 3*n)*x + (3*n^3 - 3*n)*x^2 + (n^3 - n)*x^3)*P[n - 3, x] - (3*n^3 - 3*n^2 - 6*n + (6*n^3 - 6*n^2 - 12*n)*x + (3*n^3 - 3*n^2 - 6*n)*x^2)*P[n - 2, x] + (3*n^3 - 6*n^2 - 3*n + 6 + (3*n^3 - 6*n^2 - 3*n + 6)*x)*P[n - 1, x])/(n^3 - 3*n^2 + 2*n)]];
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] = 6 + 6*x; v[3] = 18 + 36*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - n + (3*n^3 - 3*n)*x + (3*n^3 - 3*n)*x^2 + (n^3 - n)*x^3)*v[i - 3] - (3*n^3 - 3*n^2 - 6*n + (6*n^3 - 6*n^2 - 12*n)*x + (3*n^3 - 3*n^2 - 6*n)*x^2)*v[i - 2] + (3*n^3 - 6*n^2 - 3*n + 6 + (3*n^3 - 6*n^2 - 3*n + 6)*x)*v[i - 1])/(n^3 - 3*n^2 + 2*n)); 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.