Real-rooted atlas

A285061 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 10\,x + 13\,x^{2} + 4\,x^{3} - 4\,n x - 8\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1156
21264168
311274914291048
4162845348980532919472
5131293890613187417106175989420960
6115630328231170320436280313448814121548110339392
717813127821252084435964940851991510497378787121453693283038080

All 15 rows

Technical details
Node id
n-f189f7cbc9adc868 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-38c90df23fae70d9

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,       5;
  2: 1,      26,        41;
  3: 1,     127,       491,        429;
  4: 1,     628,      4534,       8980,        5329;
  5: 1,    3129,     38906,     131874,      171061,       75989;
  6: 1,   15630,    328231,    1703204,     3628031,     3448814,     1215481;
  7: 1,   78131,   2782125,   20844359,    64940851,    99151049,    73787871,    21453693;
  8: 1,  390632,  23867132,  249626648,  1069196326,  2333129944,  2750974396,  1673645480,   412820385;
  9: 1, 1953133, 207382148, 2971670276, 16854646430, 49349177494, 81997907956, 78308621908, 40150996457, 8579772325;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1-(4*n-10)*x-(8*n-13)*x^2-(4*n-4)*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*d/dx 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[((1 - (4*n - 10)*x - (8*n - 13)*x^2 - (4*n - 4)*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], 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] = ((1 - (4*n - 10)*x - (8*n - 13)*x^2 - (4*n - 4)*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(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.