Real-rooted atlas

A127648 → Veronese 2 → (θ+1)²

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 - 24\,n + 22\,n^{2} - 8\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-x^{2} + 2\,n^{2} x^{2} - n^{4} x^{2}\right)\,P_{n-4}(x) + \left(18\,x + 24\,n x - 4\,n^{2} x - 8\,n^{3} x + 2\,n^{4} x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 12\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
20
301212
40
5004545
60
7000112112
80
90000225225
100
1100000396396
120
13000000637637
140
Technical details
Node id
n-1ff264a23c480d2c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1641c37dbfc60bc3

Observed on the stored rows ?

  • Degrees: -1, 0, -1, 1, -1, 2, -1, 3, -1, 4, -1, 5, -1, 6, -1 (half affine: slope 1, intercept -1)
  • 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: 0;
  1: 1;
  2: 0;
  3: 0, 12;
  4: 0;
  5: 0,  0, 45;
  6: 0;
  7: 0,  0,  0, 112;
  8: 0;
  9: 0,  0,  0,   0, 225;
%F Formula
The row polynomials satisfy (n^4-8*n^3+22*n^2-24*n+9)*P(n,x) = -(n^4-2*n^2+1)*x^2*P(n-4,x) + (2*n^4-8*n^3-4*n^2+24*n+18)*x*P(n-2,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 0, P(3,x) = 12*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 0;
P[3, x_] := 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 2*n^2 + 1)*x^2*P[n - 4, x] + (2*n^4 - 8*n^3 - 4*n^2 + 24*n + 18)*x*P[n - 2, x])/(n^4 - 8*n^3 + 22*n^2 - 24*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 0; v[4] = 12*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^4 - 2*n^2 + 1)*x^2*v[i - 4] + (2*n^4 - 8*n^3 - 4*n^2 + 24*n + 18)*x*v[i - 2])/(n^4 - 8*n^3 + 22*n^2 - 24*n + 9)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.