Real-rooted atlas

A290319 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
212322165
31204525622311365
4168081281464811664346539585
530240178224418096488264283986658351464645
666528046164481329726420358016174777647980048151420565909025
7172972801379902084703729608883345441004088296679380876254830626408835353493178325

All 15 rows

Technical details
Node id
n-d1461fdc744b2b8a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-48fd4cbb64e3687d

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,            3;
  2:          12,           32,           21;
  3:         120,          452,          562,           231;
  4:        1680,         8128,        14648,         11664,          3465;
  5:       30240,       178224,       418096,        488264,        283986,         65835;
  6:      665280,      4616448,     13297264,      20358016,      17477764,       7980048,       1514205;
  7:    17297280,    137990208,    470372960,     888334544,    1004088296,     679380876,     254830626,      40883535;
  8:   518918400,   4675921920,  18388885248,   41231598080,   57661019744,   51508163456,   28705725936,    9126255456,   1267389585;
  9: 17643225600, 177143489280, 788879365632, 2045485318400, 3403580604096, 3769413248544, 2778780402784, 1314993093264, 362510186850, 44358635475;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n-2+(8*n-3)*x+(4*n-1)*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 - 3)*x + (4*n - 1)*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 - 3)*x + (4*n - 1)*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.