Real-rooted atlas

A008277 → θ+2 (shift +1) → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(-1 - 2\,x + 3\,x^{2} + 11\,x^{3} + 10\,x^{4} + 3\,x^{5} - n x - 4\,n x^{2} - 6\,n x^{3} - 4\,n x^{4} - n x^{5}\right)\,P_{n-2}(x) + \left(x^{2} + 4\,x^{3} + 6\,x^{4} + 4\,x^{5} + x^{6}\right)\,P_{n-2}'(x) + \left(2 + 7\,x + 8\,x^{2} + 2\,x^{3} - 2\,x^{4} - x^{5} + n x + 4\,n x^{2} + 6\,n x^{3} + 4\,n x^{4} + n x^{5}\right)\,P_{n-1}(x) + \left(-x^{2} - 4\,x^{3} - 6\,x^{4} - 4\,x^{5} - x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 3 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1358
24171536
35398452180
4674286419203988
57125755194521838775892
68195169567401308411990414037852
7928733951928557120893486954121147260132

All 15 rows

Technical details
Node id
n-65cdfe9ba679f791 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1b51648b831a15d9

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:  3,   5;
  2:  4,  17,    15;
  3:  5,  39,    84,     52;
  4:  6,  74,   286,    419,    203;
  5:  7, 125,   755,   1945,   2183,     877;
  6:  8, 195,  1695,   6740,  13084,   11990,    4140;
  7:  9, 287,  3395,  19285,  57120,   89348,   69541,   21147;
  8: 10, 404,  6244,  48062, 201796,  476714,  626630,  425473,  115975;
  9: 11, 549, 10746, 107814, 610365, 2028831, 3992189, 4538031, 2740482, 678570;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -(1+(n+2)*x+(4*n-3)*x^2+(6*n-11)*x^3+(4*n-10)*x^4+(n-3)*x^5)*P(n-2,x) + (x^2+4*x^3+6*x^4+4*x^5+x^6)*d/dx P(n-2,x) + (2+(n+7)*x+(4*n+8)*x^2+(6*n+2)*x^3+(4*n-2)*x^4+(n-1)*x^5)*P(n-1,x) - (x^2+4*x^3+6*x^4+4*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 5*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + (n + 2)*x + (4*n - 3)*x^2 + (6*n - 11)*x^3 + (4*n - 10)*x^4 + (n - 3)*x^5)*P[n - 2, x] + (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*D[P[n - 2, x], x] + (2 + (n + 7)*x + (4*n + 8)*x^2 + (6*n + 2)*x^3 + (4*n - 2)*x^4 + (n - 1)*x^5)*P[n - 1, x] - (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(1 + (n + 2)*x + (4*n - 3)*x^2 + (6*n - 11)*x^3 + (4*n - 10)*x^4 + (n - 3)*x^5)*v[i - 2] + (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*deriv(v[i - 2], x) + (2 + (n + 7)*x + (4*n + 8)*x^2 + (6*n + 2)*x^3 + (4*n - 2)*x^4 + (n - 1)*x^5)*v[i - 1] - (x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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. [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.