Real-rooted atlas

A145324 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 7\,x + 5\,x^{2} + n + 5\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13912
21275117204
360579186019894488
43604734233195099441769121176
5252042138281583939933156723310442253877632
620160410184347466615685371397959215380355730282525143472384
7181440435693644808066255832317875780532179749740620480601069993233256025840128

All 15 rows

Technical details
Node id
n-604d48e9ecd2a79d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-60de6650b202be8d

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:        3,         9;
  2:       12,        75,        117;
  3:       60,       579,       1860,        1989;
  4:      360,      4734,      23319,       50994,        41769;
  5:     2520,     42138,     281583,      939933,      1567233,       1044225;
  6:    20160,    410184,    3474666,    15685371,     39795921,      53803557,      30282525;
  7:   181440,   4356936,   44808066,   255832317,    875780532,    1797497406,    2048060106,     999323325;
  8:  1814400,  50282640,  609287292,  4216221612,  18223601049,   50378853744,   86988005082,   85771457172,   36974963025;
  9: 19958400, 627499440, 8763748452, 71359216704, 373324697631, 1301335034193, 3022401059406, 4509994237254, 3923354337327, 1515973484025;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(5*n+7)*x+(4*n+5)*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[((n + 2 + (5*n + 7)*x + (4*n + 5)*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] = ((n + 2 + (5*n + 7)*x + (4*n + 5)*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

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.