Real-rooted atlas

Veronese parent of A098172 → Veronese 3 (j=1) → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(x^{2} + 5\,x^{3} + 9\,x^{4} + 7\,x^{5} + 2\,x^{6}\right)\,P_{n-3}(x) + \left(-3\,x^{2} - 9\,x^{3} - 9\,x^{4} - 3\,x^{5}\right)\,P_{n-2}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = x\)
\(P_{3}(x) = x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2011
30011
400123
5000459
60000101121
700001222245
8000007494399
90000002811285225
10000000187255170513
110000000102405703411161
12000000005562712546832619
130000000012241590273213665913
140000000001376739395915273113365
150000000000912366955512740546130213
Technical details
Node id
n-f545410d02b04385 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0890ed59c3f00748

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 0, 1;
   4: 0, 0, 1, 2;
   5: 0, 0, 0, 4,  5;
   6: 0, 0, 0, 0, 10, 11;
   7: 0, 0, 0, 0,  1, 22, 22;
   8: 0, 0, 0, 0,  0,  7, 49,  43;
   9: 0, 0, 0, 0,  0,  0, 28, 112,  85;
  10: 0, 0, 0, 0,  0,  0,  1,  87, 255, 170;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (x^2+5*x^3+9*x^4+7*x^5+2*x^6)*P(n-3,x) - (3*x^2+9*x^3+9*x^4+3*x^5)*P(n-2,x) + (3*x+9*x^2+9*x^3+3*x^4)*P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = x, P(3,x) = x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[3, x_] := x^2;
P[n_, x_] := P[n, x] = Expand[Together[((x^2 + 5*x^3 + 9*x^4 + 7*x^5 + 2*x^6)*P[n - 3, x] - (3*x^2 + 9*x^3 + 9*x^4 + 3*x^5)*P[n - 2, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*P[n - 1, x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; v[3] = x^2; for(n = 4, N, my(i = n); v[i] = ((x^2 + 5*x^3 + 9*x^4 + 7*x^5 + 2*x^6)*v[i - 3] - (3*x^2 + 9*x^3 + 9*x^4 + 3*x^5)*v[i - 2] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*v[i - 1])/(1 + 3*x + 3*x^2 + x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.