Real-rooted atlas

A075499 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-32 + 8\,x + 3\,x^{2}\right)\,P_{n-2}(x) + \left(-16\,x + 12\,x^{2}\right)\,P_{n-2}'(x) + \left(12 - 2\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 28 + 14\,x + x^{2}\)
\(P_{1}(x) = 176 + 160\,x + 27\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02814143
1176160271364
2121617605204413541
38960194568880128065138642
468608218624144256313602660901465599
55365762498560229555270425688480492811916128472
6424345628958720362598401507430426261762136968400152187384745
7337510403392143365719449603138764807276416081016324603201344018911340126558

All 15 rows

Technical details
Node id
n-734c2ac4a375b77a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-695778c525463e73

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:         28,          14,            1;
  1:        176,         160,           27,            1;
  2:       1216,        1760,          520,           44,           1;
  3:       8960,       19456,         8880,         1280,          65,          1;
  4:      68608,      218624,       144256,        31360,        2660,         90,         1;
  5:     536576,     2498560,      2295552,       704256,       88480,       4928,       119,        1;
  6:    4243456,    28958720,     36259840,     15074304,     2626176,     213696,      8400,      152,       1;
  7:   33751040,   339214336,    571944960,    313876480,    72764160,    8101632,    460320,    13440,     189,     1;
  8:  269221888,  4003856384,   9031745536,   6430392320,  1929579520,  282078720,  21702912,   908160,   20460,   230,   1;
  9: 2150629376, 47510978560, 142908260352, 130475687936, 49708482560, 9276399616, 927429888, 52115712, 1671120, 29920, 275, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(32-8*x-3*x^2)*P(n-2,x) - (16*x-12*x^2)*d/dx P(n-2,x) + (12-2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 28 + 14*x + x^2, P(1,x) = 176 + 160*x + 27*x^2 + x^3.
%t Mathematica
P[0, x_] := 28 + 14*x + x^2;
P[1, x_] := 176 + 160*x + 27*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(32 - 8*x - 3*x^2)*P[n - 2, x] - (16*x - 12*x^2)*D[P[n - 2, x], x] + (12 - 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 28 + 14*x + x^2; v[2] = 176 + 160*x + 27*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(32 - 8*x - 3*x^2)*v[i - 2] - (16*x - 12*x^2)*deriv(v[i - 2], x) + (12 - 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], 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

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.