Real-rooted atlas

chebyshev_u → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - 5\,n^{3} x + n^{5} x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x - 2\,x^{2} + n + 4\,n x - 11\,n x^{2} - 5\,n^{2} x - 18\,n^{2} x^{2} + n^{2} x^{3} - 5\,n^{3} x - 9\,n^{3} x^{2} + 3\,n^{3} x^{3} + n^{4} x^{2} + 3\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - x^{2} - x^{3} - 10\,n x^{2} - 5\,n x^{3} - 10\,n^{2} x^{2} - 3\,n^{2} x^{3} + n^{2} x^{4} + 4\,n^{3} x^{3} + 2\,n^{3} x^{4} + 2\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
23323671
3480216128428
4516075610244002345
56280201646084000115212062
674484536153602200013824313659311
786729072422408800089856439048192281944

All 15 rows

Technical details
Node id
n-81b50c82f4bc413d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b38c31b44d930868

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:  2,    8;
  2:  3,   32,    36;
  3:  4,   80,   216,    128;
  4:  5,  160,   756,   1024,    400;
  5:  6,  280,  2016,   4608,   4000,    1152;
  6:  7,  448,  4536,  15360,  22000,   13824,    3136;
  7:  8,  672,  9072,  42240,  88000,   89856,   43904,    8192;
  8:  9,  960, 16632, 101376, 286000,  419328,  329280,  131072,  20736;
  9: 10, 1320, 28512, 219648, 800800, 1572480, 1756160, 1114112, 373248, 51200;
%F Formula
The row polynomials satisfy (n-5*n^3*x+n^5*x^2)*P(n,x) = (n+1-(5*n^3+5*n^2-4*n-4)*x+(n^5+n^4-9*n^3-18*n^2-11*n-2)*x^2+(n^5+3*n^4+3*n^3+n^2)*x^3)*P(n-1,x) + (2*x-(10*n^2+10*n+1)*x^2+(2*n^4+4*n^3-3*n^2-5*n-1)*x^3+(n^4+2*n^3+n^2)*x^4)*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[((n + 1 - (5*n^3 + 5*n^2 - 4*n - 4)*x + (n^5 + n^4 - 9*n^3 - 18*n^2 - 11*n - 2)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)*P[n - 1, x] + (2*x - (10*n^2 + 10*n + 1)*x^2 + (2*n^4 + 4*n^3 - 3*n^2 - 5*n - 1)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(n - 5*n^3*x + n^5*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] = ((n + 1 - (5*n^3 + 5*n^2 - 4*n - 4)*x + (n^5 + n^4 - 9*n^3 - 18*n^2 - 11*n - 2)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)*v[i - 1] + (2*x - (10*n^2 + 10*n + 1)*x^2 + (2*n^4 + 4*n^3 - 3*n^2 - 5*n - 1)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(n - 5*n^3*x + n^5*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

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. [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