Real-rooted atlas

image under θ+1 → (θ+1)² → θ+1

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n^{3} + 3\,n^{4} - 3\,n^{5} + n^{6}\right)\,P_{n}(x) = \left(-n^{3} x^{2} - 3\,n^{4} x^{2} - 3\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 6\,n^{2} x - 6\,n^{4} x + 2\,n^{6} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
101616
2008181
3000256256
40000625625
50000012961296
600000024012401
7000000040964096
80000000065616561
90000000001000010000
1000000000001464114641
11000000000002073620736
120000000000002856128561
1300000000000003841638416
14000000000000005062550625
Technical details
Node id
n-c0ca6457f0f11aca · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-edf6954a4cd8adfc

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: 0, 16;
  2: 0,  0, 81;
  3: 0,  0,  0, 256;
  4: 0,  0,  0,   0, 625;
  5: 0,  0,  0,   0,   0, 1296;
  6: 0,  0,  0,   0,   0,    0, 2401;
  7: 0,  0,  0,   0,   0,    0,    0, 4096;
  8: 0,  0,  0,   0,   0,    0,    0,    0, 6561;
  9: 0,  0,  0,   0,   0,    0,    0,    0,    0, 10000;
%F Formula
The row polynomials satisfy (n^6-3*n^5+3*n^4-n^3)*P(n,x) = -(n^6+3*n^5+3*n^4+n^3)*x^2*P(n-2,x) + (2*n^6-6*n^4+6*n^2-2)*x*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 3*n^5 + 3*n^4 + n^3)*x^2*P[n - 2, x] + (2*n^6 - 6*n^4 + 6*n^2 - 2)*x*P[n - 1, x])/(n^6 - 3*n^5 + 3*n^4 - n^3)]];
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] = 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 + 3*n^5 + 3*n^4 + n^3)*x^2*v[i - 2] + (2*n^6 - 6*n^4 + 6*n^2 - 2)*x*v[i - 1])/(n^6 - 3*n^5 + 3*n^4 - n^3)); 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.