Real-rooted atlas

A034839 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,n - n x + n^{3} x\right)\,P_{n}(x) = \left(6 + 12\,x - 2\,x^{2} - 6\,n - 13\,n x - n x^{2} + 2\,n^{2} x^{2} + n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x + 16\,x^{2} - 4\,x^{3} + 6\,n x^{2} - 6\,n x^{3} + 2\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2123
3167
4112316
51201536
613045480
714210528176

All 15 rows

Technical details
Node id
n-ede42eaf54339adc · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9ad63b3855519f4d

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1,  2;
  3: 1,  6;
  4: 1, 12,   3;
  5: 1, 20,  15;
  6: 1, 30,  45,   4;
  7: 1, 42, 105,  28;
  8: 1, 56, 210, 112,  5;
  9: 1, 72, 378, 336, 45;
%F Formula
The row polynomials satisfy -(6*n-6-(n^3-n)*x)*P(n,x) = -(6*n-6-(n^3-13*n+12)*x-(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (12*x-(2*n^2+6*n+16)*x^2+(2*n^2+6*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 - (n^3 - 13*n + 12)*x - (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (12*x - (2*n^2 + 6*n + 16)*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(-6*n + 6 + (n^3 - n)*x)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 6 - (n^3 - 13*n + 12)*x - (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (12*x - (2*n^2 + 6*n + 16)*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(-6*n + 6 + (n^3 - n)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.