Real-rooted atlas

A027465 → Veronese 2 (j=1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-972 + 108\,x + 972\,n - 594\,n x + 756\,n^{2} x - 3\,n^{2} x^{2} - 270\,n^{3} x + 9\,n^{3} x^{2} - 9\,n^{4} x^{2} + 3\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-2592\,x - 144\,x^{2} + 2916\,n + 972\,n x + 108\,n x^{2} + 1458\,n^{2} x + 306\,n^{2} x^{2} - 2\,n^{2} x^{3} - 810\,n^{3} x - 153\,n^{3} x^{2} - 3\,n^{3} x^{3} - 18\,n^{4} x^{2} + 9\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(5832\,x - 72\,x^{3} - 324\,n x^{2} + 36\,n x^{3} - 1620\,n^{2} x^{2} + 198\,n^{2} x^{3} - 2\,n^{2} x^{4} + 36\,n^{3} x^{3} - 4\,n^{3} x^{4} + 18\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
00
111
266
327431
410848156
54053609774
6145821601623780
751031134017011618160

All 15 rows

Technical details
Node id
n-cc29ab5cb8585e78 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-53d136ddaa9f0e1d

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     1;
  2:     6;
  3:    27,      4;
  4:   108,     48;
  5:   405,    360,     9;
  6:  1458,   2160,   162;
  7:  5103,  11340,  1701,   16;
  8: 17496,  54432, 13608,  384;
  9: 59049, 244944, 91854, 5184, 25;
%F Formula
The row polynomials satisfy (972*n-972-(270*n^3-756*n^2+594*n-108)*x+(3*n^5-9*n^4+9*n^3-3*n^2)*x^2)*P(n,x) = (2916*n-(810*n^3-1458*n^2-972*n+2592)*x+(9*n^5-18*n^4-153*n^3+306*n^2+108*n-144)*x^2+(n^5-3*n^3-2*n^2)*x^3)*P(n-1,x) + (5832*x-(1620*n^2+324*n)*x^2+(18*n^4+36*n^3+198*n^2+36*n-72)*x^3-(2*n^4+4*n^3+2*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2916*n - (810*n^3 - 1458*n^2 - 972*n + 2592)*x + (9*n^5 - 18*n^4 - 153*n^3 + 306*n^2 + 108*n - 144)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*P[n - 1, x] + (5832*x - (1620*n^2 + 324*n)*x^2 + (18*n^4 + 36*n^3 + 198*n^2 + 36*n - 72)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*D[P[n - 1, x], x])/(972*n - 972 - (270*n^3 - 756*n^2 + 594*n - 108)*x + (3*n^5 - 9*n^4 + 9*n^3 - 3*n^2)*x^2)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((2916*n - (810*n^3 - 1458*n^2 - 972*n + 2592)*x + (9*n^5 - 18*n^4 - 153*n^3 + 306*n^2 + 108*n - 144)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*v[i - 1] + (5832*x - (1620*n^2 + 324*n)*x^2 + (18*n^4 + 36*n^3 + 198*n^2 + 36*n - 72)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*deriv(v[i - 1], x))/(972*n - 972 - (270*n^3 - 756*n^2 + 594*n - 108)*x + (3*n^5 - 9*n^4 + 9*n^3 - 3*n^2)*x^2)); 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

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.