Real-rooted atlas

A161198 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 + 4\,x + 9\,x^{2} + 4\,x^{3} + 2\,n + 2\,n x - 2\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2358
3155039104
41055258194071856
594563001474214652528141920
61039586625270270402930290455815971142272
71351351351350527026510476180113277456364566145901536384256
8202702523648625110675565274999725396471075334139715153196575295823031324740608

All 15 rows

Technical details
Node id
n-f343095895214e81 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f9d3e2e230426f28

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         1;
   2:         3,          5;
   3:        15,         50,          39;
   4:       105,        525,         819,          407;
   5:       945,       6300,       14742,        14652,         5281;
   6:     10395,      86625,      270270,       402930,       290455,        81597;
   7:    135135,    1351350,     5270265,     10476180,     11327745,      6364566,      1459015;
   8:   2027025,   23648625,   110675565,    274999725,    396471075,    334139715,    153196575,     29582303;
   9:  34459425,  459459000,  2508646140,   7479992520,  13480016550,  15147667080,  10417367100,   4023193208,    669822849;
  10: 654729075, 9820936125, 61282641420, 213179786820, 461016566010, 647562767670, 593789924700, 343983019284, 114539707179, 16737058997;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(2*n+4)*x-(2*n-9)*x^2-(2*n-4)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (2*n + 4)*x - (2*n - 9)*x^2 - (2*n - 4)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 1 + (2*n + 4)*x - (2*n - 9)*x^2 - (2*n - 4)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*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, v[i])))};
T(10)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.