Real-rooted atlas

A079621 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
22448981
3192576216161000
41920768043206402514585
5230401152008640019200150036245376
632256019353601814400537600630003024494675993
75160960361267204064256015052800235200016934454886499509936

All 15 rows

Technical details
Node id
n-7baf178b1082fa11 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-baf7e4e24f192751

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:          4,           4;
  2:         24,          48,           9;
  3:        192,         576,         216,          16;
  4:       1920,        7680,        4320,         640,         25;
  5:      23040,      115200,       86400,       19200,       1500,        36;
  6:     322560,     1935360,     1814400,      537600,      63000,      3024,       49;
  7:    5160960,    36126720,    40642560,    15052800,    2352000,    169344,     5488,     64;
  8:   92897280,   743178240,   975421440,   433520640,   84672000,   8128512,   395136,   9216,    81;
  9: 1857945600, 16721510400, 25082265600, 13005619200, 3048192000, 365783040, 23708160, 829440, 14580, 100;
%F Formula
The row polynomials satisfy -(2*n-n^2*x)*P(n,x) = -(4*n^2+4*n-(2*n^3+2*n^2-4*n-4)*x-(n^2+2*n+1)*x^2)*P(n-1,x) - ((4*n+4)*x-(2*n^2+4*n+2)*x^2)*d/dx 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[(-(4*n^2 + 4*n - (2*n^3 + 2*n^2 - 4*n - 4)*x - (n^2 + 2*n + 1)*x^2)*P[n - 1, x] - ((4*n + 4)*x - (2*n^2 + 4*n + 2)*x^2)*D[P[n - 1, x], x])/(-2*n + n^2*x)]];
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] = (-(4*n^2 + 4*n - (2*n^3 + 2*n^2 - 4*n - 4)*x - (n^2 + 2*n + 1)*x^2)*v[i - 1] - ((4*n + 4)*x - (2*n^2 + 4*n + 2)*x^2)*deriv(v[i - 1], x))/(-2*n + n^2*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. [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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  7. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023