Real-rooted atlas

A008277 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 13\,x + 13\,x^{2} + 4\,x^{3} - n - 4\,n x - 5\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}(x) + \left(1 + 5\,x + 9\,x^{2} + 7\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
35151131
415679847227
5523076706412271897
62031486429961464344121517693
78777648275015220455191308237107181351
8414041847179612424411596399498541229606449592019515

All 15 rows

Technical details
Node id
n-d994d1db3c137f94 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e49234ad2c21a445

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,      67,      98,       47;
   5:     52,     307,     670,      641,      227;
   6:    203,    1486,    4299,     6146,     4344,     1215;
   7:    877,    7648,   27501,    52204,    55191,    30823,     7107;
   8:   4140,   41847,  179612,   424411,   596399,   498541,   229606,    44959;
   9:  21147,  242857, 1210098,  3417413,  5983741,  6653261,  4588620,  1795165,   305091;
  10: 115975, 1490395, 8448371, 27728366, 58078061, 80519503, 73904431, 43312372, 14710196, 2206399;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-4+(4*n-13)*x+(5*n-13)*x^2+(2*n-4)*x^3)*P(n-1,x) + (1+5*x+9*x^2+7*x^3+2*x^4)*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 - 4 + (4*n - 13)*x + (5*n - 13)*x^2 + (2*n - 4)*x^3)*P[n - 1, x] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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 - 4 + (4*n - 13)*x + (5*n - 13)*x^2 + (2*n - 4)*x^3)*v[i - 1] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + x)); 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.