Real-rooted atlas

A114687 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(6 + 12\,x + 6\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(4 - 8\,x - 84\,x^{2} - 128\,x^{3} - 56\,x^{4} + 23\,n + 143\,n x + 273\,n x^{2} + 209\,n x^{3} + 56\,n x^{4}\right)\,P_{n-1}(x) + \left(-2 - 28\,x - 128\,x^{2} - 236\,x^{3} - 190\,x^{4} - 56\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13912
2116489164
34538710719452448
419722409282165201057738816
59031276570470189210245955123129642432
6427972312498795179288035308053598872147684110974784
72079340880733830371524547540315275623979095218355118130401192085248

All 15 rows

Technical details
Node id
n-e06343e556d939fe · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-82fc5b8dd09cc2b4

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:      1;
  1:      3,        9;
  2:     11,       64,        89;
  3:     45,      387,      1071,       945;
  4:    197,     2240,      9282,     16520,      10577;
  5:    903,    12765,     70470,    189210,     245955,     123129;
  6:   4279,    72312,    498795,   1792880,    3530805,    3598872,     1476841;
  7:  20793,   408807,   3383037,  15245475,   40315275,   62397909,    52183551,    18130401;
  8: 103049,  2310496,  22303652, 120888880,  401713214,  836436160,  1063414772,   752959024,   226739489;
  9: 518859, 13065489, 144127404, 913043124, 3655771434, 9579918222, 16403236668, 17664819012, 10835780643, 2878666857;
%F Formula
The row polynomials satisfy (3*n+6+(6*n+12)*x+(3*n+6)*x^2)*P(n,x) = (23*n+4+(143*n-8)*x+(273*n-84)*x^2+(209*n-128)*x^3+(56*n-56)*x^4)*P(n-1,x) - (2+28*x+128*x^2+236*x^3+190*x^4+56*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((23*n + 4 + (143*n - 8)*x + (273*n - 84)*x^2 + (209*n - 128)*x^3 + (56*n - 56)*x^4)*P[n - 1, x] - (2 + 28*x + 128*x^2 + 236*x^3 + 190*x^4 + 56*x^5)*D[P[n - 1, x], x])/(3*n + 6 + (6*n + 12)*x + (3*n + 6)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((23*n + 4 + (143*n - 8)*x + (273*n - 84)*x^2 + (209*n - 128)*x^3 + (56*n - 56)*x^4)*v[i - 1] - (2 + 28*x + 128*x^2 + 236*x^3 + 190*x^4 + 56*x^5)*deriv(v[i - 1], x))/(3*n + 6 + (6*n + 12)*x + (3*n + 6)*x^2)); 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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023