Real-rooted atlas

A114687 → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2118120
34551151112
4197304146241672
5903176511903303514224
64279101288835352064548127456
72079357799620133251587151141631183040

All 15 rows

Technical details
Node id
n-2a5188b3bacc3f29 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c8aa5aaa0c2f2cd0

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,       1;
  2:     11,       8,       1;
  3:     45,      51,      15,       1;
  4:    197,     304,     146,      24,      1;
  5:    903,    1765,    1190,     330,     35,      1;
  6:   4279,   10128,    8835,    3520,    645,     48,     1;
  7:  20793,   57799,   62013,   32515,   8715,   1141,    63,    1;
  8: 103049,  329024,  419524,  274064,  98014,  19040,  1876,   80,  1;
  9: 518859, 1871145, 2766060, 2169300, 977130, 256158, 37884, 2916, 99, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (5*n+4+3*n*x)*P(n-1,x) + (2-2*x^2)*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[((5*n + 4 + 3*n*x)*P[n - 1, x] + (2 - 2*x^2)*D[P[n - 1, x], x])/(n + 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] = ((5*n + 4 + 3*n*x)*v[i - 1] + (2 - 2*x^2)*deriv(v[i - 1], x))/(n + 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

Same fingerprint ?

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