Real-rooted atlas

A008292 → 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

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-3\,x - 10\,x^{2} - 11\,x^{3} - 4\,x^{4} + n + 5\,n x + 9\,n x^{2} + 7\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(-x - 5\,x^{2} - 9\,x^{3} - 7\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
36181337
42410815875365
5120720159015305414501
67205400159602322016622468366605
7504045360168000327600354606201978472931149877
8403204233601884960461160066936245762988272487854583522687565

All 15 rows

Technical details
Node id
n-77c7c07dd77d9ae1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d4e4e266d9f48188

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:       2,        3;
   3:       6,       18,        13;
   4:      24,      108,       158,         75;
   5:     120,      720,      1590,       1530,        541;
   6:     720,     5400,     15960,      23220,      16622,       4683;
   7:    5040,    45360,    168000,     327600,     354606,     201978,      47293;
   8:   40320,   423360,   1884960,    4611600,    6693624,    5762988,    2724878,     545835;
   9:  362880,  4354560,  22649760,   66679200,  121491720,  140268240,  100203270,   40492890,   7087261;
  10: 3628800, 48988800, 291513600, 1003363200, 2200973040, 3190509000, 3055965720, 1864917540, 657944942, 102247563;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n+(5*n-3)*x+(9*n-10)*x^2+(7*n-11)*x^3+(2*n-4)*x^4)*P(n-1,x) - (x+5*x^2+9*x^3+7*x^4+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[((n + (5*n - 3)*x + (9*n - 10)*x^2 + (7*n - 11)*x^3 + (2*n - 4)*x^4)*P[n - 1, x] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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] = ((n + (5*n - 3)*x + (9*n - 10)*x^2 + (7*n - 11)*x^3 + (2*n - 4)*x^4)*v[i - 1] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*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, v[i])))};
T(10)

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. [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.