Real-rooted atlas

A089627 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-17 - 8\,x + 9\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-28 - 30\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
3527
413619
54936691
616114030331
7581630210201441
82045256210501405797
9739310696546011207024739
102668943416259566720630103411
1197285176490122220394805670252441397
123555657102705563802125203927027721876777
131305745285054024977701108800256410277209248047909
1448085451138410011021010555984015315302162161201234533253
1517760965453334704807803027207180873873015495481321323432148803487
Technical details
Node id
n-4e15a6be7981b1f4 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bd9a36d3d6b9d933

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 -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:     1;
   3:     5,     2;
   4:    13,     6;
   5:    49,    36,     6;
   6:   161,   140,    30;
   7:   581,   630,   210,   20;
   8:  2045,  2562,  1050,  140;
   9:  7393, 10696,  5460, 1120,  70;
  10: 26689, 43416, 25956, 6720, 630;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (9*n-17+(4*n-8)*x)*P(n-1,x) - (28+30*x+8*x^2)*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[((9*n - 17 + (4*n - 8)*x)*P[n - 1, x] - (28 + 30*x + 8*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = ((9*n - 17 + (4*n - 8)*x)*v[i - 1] - (28 + 30*x + 8*x^2)*deriv(v[i - 1], x))/(n - 1)); 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. [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