Real-rooted atlas

A145324 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
2126854134
3605328823841858
436043921127710144300029173
525203938413823119129611610025920513451
620160385632172614632505763021100137764824696010028222
7181440411580822476690538048166782210047201904172236962580480215406934

All 15 rows

Technical details
Node id
n-3cd852da1a767dd6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6449e3f2a5084b2d

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:        3,         8;
  2:       12,        68,         54;
  3:       60,       532,        882,         384;
  4:      360,      4392,      11277,       10144,        3000;
  5:     2520,     39384,     138231,      191296,      116100,       25920;
  6:    20160,    385632,    1726146,     3250576,     3021100,     1377648,      246960;
  7:   181440,   4115808,   22476690,    53804816,    67822100,    47201904,    17223696,     2580480;
  8:  1814400,  47689920,  308112012,   897675200,  1434851225,  1350993456,   750460284,   228271104,   29393280;
  9: 19958400, 597165120, 4462255332, 15351974080, 29809538475, 35522785656, 26643585164, 12312912384, 3212382240, 362880000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(4*n^3+8*n^2+4*n)*x-(9*n^3+9*n^2)*x^2)*P(n-2,x) + ((4*n^3+8*n^2+4*n)*x^2+(4*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+1+(4*n^2+4*n)*x)*P(n-1,x) - (4*n^2+4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x - (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((4*n^3 + 8*n^2 + 4*n)*x^2 + (4*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*P[n - 1, x] - (4*n^2 + 4*n)*x^2*D[P[n - 1, x], x])/(n)]];
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] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x - (9*n^3 + 9*n^2)*x^2)*v[i - 2] + ((4*n^3 + 8*n^2 + 4*n)*x^2 + (4*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*v[i - 1] - (4*n^2 + 4*n)*x^2*deriv(v[i - 1], x))/(n)); 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. [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.