Real-rooted atlas

chebyshev_t → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-9 + 6\,x + 9\,n - 28\,n x + 33\,n^{2} x - 4\,n^{2} x^{2} - 11\,n^{3} x + 8\,n^{3} x^{2} - 5\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-9 - 30\,x - 20\,x^{2} - 4\,x^{3} + 9\,n + 8\,n x + n x^{2} + 33\,n^{2} x + 34\,n^{2} x^{2} + 7\,n^{2} x^{3} - 11\,n^{3} x - 11\,n^{3} x^{2} - n^{3} x^{3} - 5\,n^{4} x^{2} - 3\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(18\,x + 35\,x^{2} + 21\,x^{3} + 4\,x^{4} + 22\,n x^{2} + 19\,n x^{3} + 4\,n x^{4} - 22\,n^{2} x^{2} - 17\,n^{2} x^{3} - 3\,n^{2} x^{4} - 4\,n^{3} x^{3} - 2\,n^{3} x^{4} + 2\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
21161835
313610864209
41643605122001137
51100900224020005765817
6114418907168108006912156828483
711963528188164200044352219524096134941

All 15 rows

Technical details
Node id
n-c443535979497fee · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ef846d88422ec39f

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: 1,   4;
  2: 1,  16,   18;
  3: 1,  36,  108,    64;
  4: 1,  64,  360,   512,    200;
  5: 1, 100,  900,  2240,   2000,    576;
  6: 1, 144, 1890,  7168,  10800,   6912,   1568;
  7: 1, 196, 3528, 18816,  42000,  44352,  21952,   4096;
  8: 1, 256, 6048, 43008, 132000, 202752, 163072,  65536,  10368;
  9: 1, 324, 9720, 88704, 356400, 741312, 856128, 552960, 186624, 25600;
%F Formula
The row polynomials satisfy (9*n-9-(11*n^3-33*n^2+28*n-6)*x+(n^5-5*n^4+8*n^3-4*n^2)*x^2)*P(n,x) = (9*n-9-(11*n^3-33*n^2-8*n+30)*x+(n^5-5*n^4-11*n^3+34*n^2+n-20)*x^2+(n^5-3*n^4-n^3+7*n^2-4)*x^3)*P(n-1,x) + (18*x-(22*n^2-22*n-35)*x^2+(2*n^4-4*n^3-17*n^2+19*n+21)*x^3+(n^4-2*n^3-3*n^2+4*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 9 - (11*n^3 - 33*n^2 - 8*n + 30)*x + (n^5 - 5*n^4 - 11*n^3 + 34*n^2 + n - 20)*x^2 + (n^5 - 3*n^4 - n^3 + 7*n^2 - 4)*x^3)*P[n - 1, x] + (18*x - (22*n^2 - 22*n - 35)*x^2 + (2*n^4 - 4*n^3 - 17*n^2 + 19*n + 21)*x^3 + (n^4 - 2*n^3 - 3*n^2 + 4*n + 4)*x^4)*D[P[n - 1, x], x])/(9*n - 9 - (11*n^3 - 33*n^2 + 28*n - 6)*x + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*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; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n - 9 - (11*n^3 - 33*n^2 - 8*n + 30)*x + (n^5 - 5*n^4 - 11*n^3 + 34*n^2 + n - 20)*x^2 + (n^5 - 3*n^4 - n^3 + 7*n^2 - 4)*x^3)*v[i - 1] + (18*x - (22*n^2 - 22*n - 35)*x^2 + (2*n^4 - 4*n^3 - 17*n^2 + 19*n + 21)*x^3 + (n^4 - 2*n^3 - 3*n^2 + 4*n + 4)*x^4)*deriv(v[i - 1], x))/(9*n - 9 - (11*n^3 - 33*n^2 + 28*n - 6)*x + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023