Real-rooted atlas

A085881 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 3\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-1 - 3\,x - 3\,x^{2} - x^{3} + 2\,n + 3\,n x + n x^{2} + 6\,n^{2} x + 9\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
23242754
315180405240840
4105168056706720262516800
5945189008505015120011812534020408240
610395249480140332533264003898125224532050935511642400
713513537837802554051575675600118243125102162060463513058648640380540160

All 15 rows

Technical details
Node id
n-3dd45cfabd20c8dc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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: 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,          4;
  2:        3,         24,          27;
  3:       15,        180,         405,         240;
  4:      105,       1680,        5670,        6720,         2625;
  5:      945,      18900,       85050,      151200,       118125,        34020;
  6:    10395,     249480,     1403325,     3326400,      3898125,      2245320,       509355;
  7:   135135,    3783780,    25540515,    75675600,    118243125,    102162060,     46351305,     8648640;
  8:  2027025,   64864800,   510810300,  1816214400,   3547293750,   4086482400,   2781078300,  1037836800,   164189025;
  9: 34459425, 1240539300, 11164853700, 46313467200, 108547188750, 156307951800, 141834993300, 79394515200, 25120920825, 3445942500;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (2*n-1+(6*n^2+3*n-3)*x+(2*n^3+9*n^2+n-3)*x^2+(2*n^3+3*n^2-1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (6*n^2 + 3*n - 3)*x + (2*n^3 + 9*n^2 + n - 3)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n - 1 + (6*n^2 + 3*n - 3)*x + (2*n^3 + 9*n^2 + n - 3)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + 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

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.