Real-rooted atlas

A292219 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - x + n x\right)\,P_{n}(x) = \left(4 - 2\,x - 8\,n - 6\,n x + n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2628
360403103
484084012641810
515120201604536288540109
63326405544001663201584055061069756
7864864017297280648648082368042900936733299923
825945920060540480027243216043243200300300098280147081183642118

All 15 rows

Technical details
Node id
n-c05cf5b45147fec7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-692d0ae5c5a32a15

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            1;
   2:            6,             2;
   3:           60,            40,            3;
   4:          840,           840,          126,            4;
   5:        15120,         20160,         4536,          288,           5;
   6:       332640,        554400,       166320,        15840,         550,         6;
   7:      8648640,      17297280,      6486480,       823680,       42900,       936,        7;
   8:    259459200,     605404800,    272432160,     43243200,     3003000,     98280,     1470,      8;
   9:   8821612800,   23524300800,  12350257920,   2352430080,   204204000,   8910720,   199920,   2176,    9;
  10: 335221286400, 1005663859200, 603398315520, 134088514560, 13967553600, 761866560, 22790880, 372096, 3078, 10;
%F Formula
The row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(8*n-4-(4*n^2-6*n-2)*x-n*x^2)*P(n-1,x) - (8*x-4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 4 - (4*n^2 - 6*n - 2)*x - n*x^2)*P[n - 1, x] - (8*x - 4*n*x^2)*D[P[n - 1, x], x])/(-2 + (n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n - 4 - (4*n^2 - 6*n - 2)*x - n*x^2)*v[i - 1] - (8*x - 4*n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 1)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.