Real-rooted atlas

A062145 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 4\,x - 2\,x^{2} + n + 2\,n x + n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-4\,x^{2} - 8\,x^{3} - 4\,x^{4} + 2\,n + 4\,n x - 4\,n x^{2} - 12\,n x^{3} - 6\,n x^{4} + 4\,n^{2} + 17\,n^{2} x + 26\,n^{2} x^{2} + 17\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-x - 5\,x^{2} - 9\,x^{3} - 7\,x^{4} - 2\,x^{5} - 2\,n x - 10\,n x^{2} - 18\,n x^{3} - 14\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
25914
3217261154
4844206933771574
5330216052565632224115619
612871039533330530104180513073152900
7500548048190905401720472065293640755171486900
8194482162161024023267767541741703878490198852543390514412452

All 15 rows

Technical details
Node id
n-42ac6f7d24f92f67 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7a903cd6634e770d

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:      5,       9;
   3:     21,      72,       61;
   4:     84,     420,      693,      377;
   5:    330,    2160,     5256,     5632,      2241;
   6:   1287,   10395,    33330,    53010,     41805,     13073;
   7:   5005,   48048,   190905,   401720,    472065,    293640,     75517;
   8:  19448,  216216,  1024023,  2677675,   4174170,   3878490,   1988525,    433905;
   9:  75582,  954720,  5247424, 16388736,  31807230,  39275600,  30128280,  13125120,  2485825;
  10: 293930, 4157010, 26000208, 94379376, 219092328, 337264200, 344236620, 224618940, 85015125, 14218905;
%F Formula
The row polynomials satisfy (n^2+n-2+(2*n^2+2*n-4)*x+(n^2+n-2)*x^2)*P(n,x) = (4*n^2+2*n+(17*n^2+4*n)*x+(26*n^2-4*n-4)*x^2+(17*n^2-12*n-8)*x^3+(4*n^2-6*n-4)*x^4)*P(n-1,x) - ((2*n+1)*x+(10*n+5)*x^2+(18*n+9)*x^3+(14*n+7)*x^4+(4*n+2)*x^5)*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[((4*n^2 + 2*n + (17*n^2 + 4*n)*x + (26*n^2 - 4*n - 4)*x^2 + (17*n^2 - 12*n - 8)*x^3 + (4*n^2 - 6*n - 4)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (10*n + 5)*x^2 + (18*n + 9)*x^3 + (14*n + 7)*x^4 + (4*n + 2)*x^5)*D[P[n - 1, x], x])/(n^2 + n - 2 + (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)]];
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] = ((4*n^2 + 2*n + (17*n^2 + 4*n)*x + (26*n^2 - 4*n - 4)*x^2 + (17*n^2 - 12*n - 8)*x^3 + (4*n^2 - 6*n - 4)*x^4)*v[i - 1] - ((2*n + 1)*x + (10*n + 5)*x^2 + (18*n + 9)*x^3 + (14*n + 7)*x^4 + (4*n + 2)*x^5)*deriv(v[i - 1], x))/(n^2 + n - 2 + (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.