Real-rooted atlas

A227450 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 5\,x^{2} - x^{4} + n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4}\right)\,P_{n-1}(x) + \left(-x^{2} - 3\,x^{3} - 3\,x^{4} - x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
14610
28282258
3169617294378
43228887210684542714
564800360073606924243021178
612821121312039520614484727614214177818
7256537643904181440412496520352340428899181594170

All 15 rows

Technical details
Node id
n-e8f2ea6fc35437d7 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6319fdf993d134a2

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:    2;
  1:    4,     6;
  2:    8,    28,     22;
  3:   16,    96,    172,      94;
  4:   32,   288,    872,    1068,      454;
  5:   64,   800,   3600,    7360,     6924,     2430;
  6:  128,  2112,  13120,   39520,    61448,    47276,    14214;
  7:  256,  5376,  43904,  181440,   412496,   520352,   340428,    89918;
  8:  512, 13312, 137984,  746368,  2314592,  4241440,  4520104,  2582252,   610182;
  9: 1024, 32256, 413184, 2827776, 11428032, 28416864, 43747216, 40489344, 20589580, 4412798;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+(n+6)*x+(3*n+5)*x^2+3*n*x^3+(n-1)*x^4)*P(n-1,x) - (x^2+3*x^3+3*x^4+x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (n + 6)*x + (3*n + 5)*x^2 + 3*n*x^3 + (n - 1)*x^4)*P[n - 1, x] - (x^2 + 3*x^3 + 3*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (n + 6)*x + (3*n + 5)*x^2 + 3*n*x^3 + (n - 1)*x^4)*v[i - 1] - (x^2 + 3*x^3 + 3*x^4 + x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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.