Real-rooted atlas

A145597 → γ → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + n^{2}\right)\,P_{n}(x) = \left(4 - 8\,x + 12\,n + 4\,n x + 5\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-12 - 28\,x - 16\,x^{2} - 6\,n - 14\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
29312
3251540
469639141
518923163483
6518798336281680
71422264615122525832

All 15 rows

Technical details
Node id
n-636394250e7e19d6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7c82afb7cf1d521a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:     9,     3;
  3:    25,    15;
  4:    69,    63,     9;
  5:   189,   231,    63;
  6:   518,   798,   336,   28;
  7:  1422,  2646,  1512,  252;
  8:  3915,  8550,  6210, 1620,  90;
  9: 10813, 27126, 23958, 8580, 990;
%F Formula
The row polynomials satisfy (n^2+6*n)*P(n,x) = (5*n^2+12*n+4+(4*n^2+4*n-8)*x)*P(n-1,x) - (6*n+12+(14*n+28)*x+(8*n+16)*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^2 + 12*n + 4 + (4*n^2 + 4*n - 8)*x)*P[n - 1, x] - (6*n + 12 + (14*n + 28)*x + (8*n + 16)*x^2)*D[P[n - 1, x], x])/(n^2 + 6*n)]];
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^2 + 12*n + 4 + (4*n^2 + 4*n - 8)*x)*v[i - 1] - (6*n + 12 + (14*n + 28)*x + (8*n + 16)*x^2)*deriv(v[i - 1], x))/(n^2 + 6*n)); 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

Candidate equalities ?

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