Real-rooted atlas

A075505 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-200 - 20\,x - x^{2}\right)\,P_{n-2}(x) + \left(-100\,x - 10\,x^{2}\right)\,P_{n-2}'(x) + \left(30 + 2\,x\right)\,P_{n-1}(x) + \left(10\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 30 + 2\,x\)
\(P_{1}(x) = 700 + 120\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
030232
17001203823
2150005000300420304
3310000180000195006005510105
4630000060200001050000560001050613427056
512700000019320000051030000420000013300016807375564687
6255000000060500000002331000000278040000132300002772002520811222549728
7511000000001866000000001023150000001701000000011413500003528000052500036009358202158609

All 15 rows

Technical details
Node id
n-788e4697181967e9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9adf2978aaed3e00

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             30,               2;
  1:            700,             120,               3;
  2:          15000,            5000,             300,              4;
  3:         310000,          180000,           19500,            600,             5;
  4:        6300000,         6020000,         1050000,          56000,          1050,            6;
  5:      127000000,       193200000,        51030000,        4200000,        133000,         1680,           7;
  6:     2550000000,      6050000000,      2331000000,      278040000,      13230000,       277200,        2520,         8;
  7:    51100000000,    186600000000,    102315000000,    17010000000,    1141350000,     35280000,      525000,      3600,       9;
  8:  1023000000000,   5700200000000,   4372500000000,   986920000000,   89743500000,   3839220000,    83160000,    924000,    4950,   10;
  9: 20470000000000, 173052000000000, 183450300000000, 55176000000000, 6618260000000, 376437600000, 11131890000, 178200000, 1534500, 6600, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(200+20*x+x^2)*P(n-2,x) - (100*x+10*x^2)*d/dx P(n-2,x) + (30+2*x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 30 + 2*x, P(1,x) = 700 + 120*x + 3*x^2.
%t Mathematica
P[0, x_] := 30 + 2*x;
P[1, x_] := 700 + 120*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(200 + 20*x + x^2)*P[n - 2, x] - (100*x + 10*x^2)*D[P[n - 2, x], x] + (30 + 2*x)*P[n - 1, x] + 10*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] = 30 + 2*x; v[2] = 700 + 120*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(200 + 20*x + x^2)*v[i - 2] - (100*x + 10*x^2)*deriv(v[i - 2], x) + (30 + 2*x)*v[i - 1] + 10*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.