Real-rooted atlas

A008277 → θ+1 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-2 - 13\,x - 34\,x^{2} - 43\,x^{3} - 26\,x^{4} - 6\,x^{5} + n x + 5\,n x^{2} + 9\,n x^{3} + 7\,n x^{4} + 2\,n x^{5}\right)\,P_{n-2}(x) + \left(-x - 6\,x^{2} - 14\,x^{3} - 16\,x^{4} - 9\,x^{5} - 2\,x^{6}\right)\,P_{n-2}'(x) + \left(3 + 13\,x + 19\,x^{2} + 11\,x^{3} + 2\,x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2358
37262255
4159517599384
53130492111144712841
663917408879527098238622504
712726941666846472657734626212867190863
8255783564965244029487471538491311337736811728064

All 15 rows

Technical details
Node id
n-81e8b2a64b21a60e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ac114d6c33f8e4e9

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:    7,    26,     22;
   4:   15,    95,    175,      99;
   5:   31,   304,    921,    1114,      471;
   6:   63,   917,   4088,    7952,     7098,     2386;
   7:  127,  2694,  16668,   46472,    65773,    46262,    12867;
   8:  255,  7835,  64965,  244029,   487471,   538491,   311337,    73681;
   9:  511, 22748, 247243, 1204466,  3188230,  4920836,  4437793,  2172338,   446620;
  10: 1023, 66209, 930094, 5729658, 19322215, 39097685, 48913392, 37155536, 15738811, 2856457;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(2-(n-13)*x-(5*n-34)*x^2-(9*n-43)*x^3-(7*n-26)*x^4-(2*n-6)*x^5)*P(n-2,x) - (x+6*x^2+14*x^3+16*x^4+9*x^5+2*x^6)*d/dx P(n-2,x) + (3-(n-13)*x-(3*n-19)*x^2-(3*n-11)*x^3-(n-2)*x^4)*P(n-1,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 5*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 - (n - 13)*x - (5*n - 34)*x^2 - (9*n - 43)*x^3 - (7*n - 26)*x^4 - (2*n - 6)*x^5)*P[n - 2, x] - (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 2, x], x] + (3 - (n - 13)*x - (3*n - 19)*x^2 - (3*n - 11)*x^3 - (n - 2)*x^4)*P[n - 1, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + 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; v[2] = 3 + 5*x; for(n = 3, N, my(i = n); v[i] = (-(2 - (n - 13)*x - (5*n - 34)*x^2 - (9*n - 43)*x^3 - (7*n - 26)*x^4 - (2*n - 6)*x^5)*v[i - 2] - (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 2], x) + (3 - (n - 13)*x - (3*n - 19)*x^2 - (3*n - 11)*x^3 - (n - 2)*x^4)*v[i - 1] + (x + 4*x^2 + 6*x^3 + 4*x^4 + 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

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.