Real-rooted atlas

A114656 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - n^{2} + 2\,n^{2} x - 4\,n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(-8\,x - 8\,x^{2} + n + 20\,n x + 20\,n x^{2} - n^{2} - 10\,n^{2} x - 8\,n^{2} x^{2} - 4\,n^{3} x - 16\,n^{3} x^{2} + 2\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(x - 4\,x^{3} - 2\,n x + 8\,n x^{3} + 4\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2189
31243661
4148216128393
518072012804002481
61120180064006000115215473
711683780224004200024192313695677
81224705662720196000225792878088192587793

All 15 rows

Technical details
Node id
n-b28735f6d6418132 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e9eb093223c2e083

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 -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,   8;
   3: 1,  24,    36;
   4: 1,  48,   216,    128;
   5: 1,  80,   720,   1280,     400;
   6: 1, 120,  1800,   6400,    6000,    1152;
   7: 1, 168,  3780,  22400,   42000,   24192,    3136;
   8: 1, 224,  7056,  62720,  196000,  225792,   87808,    8192;
   9: 1, 288, 12096, 150528,  705600, 1354752, 1053696,  294912,  20736;
  10: 1, 360, 19440, 322560, 2116800, 6096384, 7902720, 4423680, 933120, 51200;
%F Formula
The row polynomials satisfy -(n^2-n-(2*n^4-4*n^3+2*n^2)*x)*P(n,x) = -(n^2-n-(2*n^4-4*n^3-10*n^2+20*n-8)*x-(12*n^4-16*n^3-8*n^2+20*n-8)*x^2)*P(n-1,x) - ((2*n-1)*x-4*n^3*x^2+(8*n^3-8*n+4)*x^3)*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[(-(n^2 - n - (2*n^4 - 4*n^3 - 10*n^2 + 20*n - 8)*x - (12*n^4 - 16*n^3 - 8*n^2 + 20*n - 8)*x^2)*P[n - 1, x] - ((2*n - 1)*x - 4*n^3*x^2 + (8*n^3 - 8*n + 4)*x^3)*D[P[n - 1, x], x])/(-n^2 + n + (2*n^4 - 4*n^3 + 2*n^2)*x)]];
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] = (-(n^2 - n - (2*n^4 - 4*n^3 - 10*n^2 + 20*n - 8)*x - (12*n^4 - 16*n^3 - 8*n^2 + 20*n - 8)*x^2)*v[i - 1] - ((2*n - 1)*x - 4*n^3*x^2 + (8*n^3 - 8*n + 4)*x^3)*deriv(v[i - 1], x))/(-n^2 + n + (2*n^4 - 4*n^3 + 2*n^2)*x)); 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