Real-rooted atlas

A099040 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2 + 14\,x + 28\,x^{2} + 22\,x^{3} + 6\,x^{4} - 2\,n - 10\,n x - 20\,n x^{2} - 18\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 18\,x^{2} + 30\,x^{3} + 22\,x^{4} + 6\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1044
2041620
300245680
40012120188320
5000965126081216
600032576201619204544
7000032028807488595216640
8000080224012960266881819260160
90000096012800541449216054976215040
1000000192768064640214272310464164608761856
1100000026884928029926481331210251524890882678784
120000004482419227552012992002986560332966414437769359360
1300000007168172032139776053657601067520010665984423833632522240
1400000001024716801053696659456021294080373125123376640012382208112476160
Technical details
Node id
n-bf59793c421d5843 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-233371af38a31cb8

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: 0, 4;
  2: 0, 4, 16;
  3: 0, 0, 24,  56;
  4: 0, 0, 12, 120, 188;
  5: 0, 0,  0,  96, 512,  608;
  6: 0, 0,  0,  32, 576, 2016,  1920;
  7: 0, 0,  0,   0, 320, 2880,  7488,  5952;
  8: 0, 0,  0,   0,  80, 2240, 12960, 26688, 18192;
  9: 0, 0,  0,   0,   0,  960, 12800, 54144, 92160, 54976;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(2*n-2+(10*n-14)*x+(20*n-28)*x^2+(18*n-22)*x^3+(6*n-6)*x^4)*P(n-1,x) + (4*x+18*x^2+30*x^3+22*x^4+6*x^5)*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[(-(2*n - 2 + (10*n - 14)*x + (20*n - 28)*x^2 + (18*n - 22)*x^3 + (6*n - 6)*x^4)*P[n - 1, x] + (4*x + 18*x^2 + 30*x^3 + 22*x^4 + 6*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
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] = (-(2*n - 2 + (10*n - 14)*x + (20*n - 28)*x^2 + (18*n - 22)*x^3 + (6*n - 6)*x^4)*v[i - 1] + (4*x + 18*x^2 + 30*x^3 + 22*x^4 + 6*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.