Real-rooted atlas

A008278 → 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

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-2 - 13\,x - 28\,x^{2} - 25\,x^{3} - 8\,x^{4} + 2\,n + 10\,n x + 18\,n x^{2} + 14\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 7\,x - 19\,x^{2} - 25\,x^{3} - 16\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
35151131
4156810149233
5523177157062572047
62031559473771115268153920417
787781383117663068710044214910299226711
8414045093208826532724808067728411361099759052764265

All 15 rows

Technical details
Node id
n-e6038c739ca0a01b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a11451714fced61e

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:      5,      15,       11;
   4:     15,      68,      101,       49;
   5:     52,     317,      715,      706,      257;
   6:    203,    1559,     4737,     7111,     5268,      1539;
   7:    877,    8138,    31176,    63068,    71004,     42149,     10299;
   8:   4140,   45093,   208826,   532724,   808067,    728411,    361099,    75905;
   9:  21147,  264671,  1439145,  4438642,  8489401,  10306162,   7752138,  3301796,   609441;
  10: 115975, 1640930, 10254791, 37139185, 85873202, 131415497, 133059970, 85924298, 32100422, 5284451;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2*n-2+(10*n-13)*x+(18*n-28)*x^2+(14*n-25)*x^3+(4*n-8)*x^4)*P(n-1,x) - (1+7*x+19*x^2+25*x^3+16*x^4+4*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[((2*n - 2 + (10*n - 13)*x + (18*n - 28)*x^2 + (14*n - 25)*x^3 + (4*n - 8)*x^4)*P[n - 1, x] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 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; for(n = 2, N, my(i = n); v[i] = ((2*n - 2 + (10*n - 13)*x + (18*n - 28)*x^2 + (14*n - 25)*x^3 + (4*n - 8)*x^4)*v[i - 1] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 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

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.