Real-rooted atlas

A290319 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
2125660128
312072813848403072
416801235232480366721512098304
530240259296856384137465610789443326403932160
66652806490752255821445250841659477824353706248648640188743680
717297280188717952859858304213268313631216759042703970944128598336025945920010569646080

All 15 rows

Technical details
Node id
n-3e3a66f1d1f268f6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-438412a981173c5d

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           2,            6;
  2:          12,           56,            60;
  3:         120,          728,          1384,           840;
  4:        1680,        12352,         32480,         36672,         15120;
  5:       30240,       259296,        856384,       1374656,       1078944,         332640;
  6:      665280,      6490752,      25582144,      52508416,      59477824,       35370624,        8648640;
  7:    17297280,    188717952,     859858304,    2132683136,    3121675904,     2703970944,     1285983360,     259459200;
  8:   518918400,   6249646080,   32212159488,   93215676416,  166161503744,   187256109056,   130514512896,   51507210240,    8821612800;
  9: 17643225600, 232206865920, 1332699973632, 4393395058688, 9191686831104, 12680844850176, 11553225582592, 6710796638208, 2257208824320, 335221286400;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n-2+8*n*x+(4*n+2)*x^2)*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 + 8*n*x + (4*n + 2)*x^2)*P[n - 1, x])/(1 + 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 + 8*n*x + (4*n + 2)*x^2)*v[i - 1])/(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

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. [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.