Real-rooted atlas

A026729 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-40 - 128\,x + 40\,n + 48\,n x - 128\,n x^{2} + 80\,n^{2} x + 64\,n^{2} x^{2} + 64\,n^{3} x^{2}\right)\,P_{n}(x) = \left(200 + 640\,x + 384\,x^{2} - 80\,n + 464\,n x + 1600\,n x^{2} + 1024\,n x^{3} - 240\,n^{2} x + 64\,n^{2} x^{2} + 512\,n^{2} x^{3} - 128\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 180\,x + 368\,x^{2} + 192\,x^{3} + 100\,n x + 640\,n x^{2} + 1088\,n x^{3} + 512\,n x^{4} + 80\,n^{2} x^{2} + 384\,n^{2} x^{3} + 256\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2189
32244874
4356240256555
55120768179212803965
682402064742411520614427400
713464499224320588806758428672184925
82187211280693762291204116483727361310721226125

All 15 rows

Technical details
Node id
n-ac6d2c2a4f0a2ec1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-33e90ed15bb6d064

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:  2,   24,    48;
   4:  3,   56,   240,    256;
   5:  5,  120,   768,   1792,    1280;
   6:  8,  240,  2064,   7424,   11520,    6144;
   7: 13,  464,  4992,  24320,   58880,   67584,    28672;
   8: 21,  872, 11280,  69376,  229120,  411648,   372736,   131072;
   9: 34, 1608, 24288, 180480,  756480, 1861632,  2637824,  1966080,   589824;
  10: 55, 2920, 50448, 439552, 2234880, 7004160, 13619200, 15859712, 10027008, 2621440;
%F Formula
The row polynomials satisfy (40*n-40+(80*n^2+48*n-128)*x+(64*n^3+64*n^2-128*n)*x^2)*P(n,x) = -(80*n-200+(240*n^2-464*n-640)*x+(128*n^3-64*n^2-1600*n-384)*x^2-(512*n^2+1024*n)*x^3)*P(n-1,x) + (25+(100*n+180)*x+(80*n^2+640*n+368)*x^2+(384*n^2+1088*n+192)*x^3+(256*n^2+512*n)*x^4)*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[(-(80*n - 200 + (240*n^2 - 464*n - 640)*x + (128*n^3 - 64*n^2 - 1600*n - 384)*x^2 - (512*n^2 + 1024*n)*x^3)*P[n - 1, x] + (25 + (100*n + 180)*x + (80*n^2 + 640*n + 368)*x^2 + (384*n^2 + 1088*n + 192)*x^3 + (256*n^2 + 512*n)*x^4)*D[P[n - 1, x], x])/(40*n - 40 + (80*n^2 + 48*n - 128)*x + (64*n^3 + 64*n^2 - 128*n)*x^2)]];
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] = (-(80*n - 200 + (240*n^2 - 464*n - 640)*x + (128*n^3 - 64*n^2 - 1600*n - 384)*x^2 - (512*n^2 + 1024*n)*x^3)*v[i - 1] + (25 + (100*n + 180)*x + (80*n^2 + 640*n + 368)*x^2 + (384*n^2 + 1088*n + 192)*x^3 + (256*n^2 + 512*n)*x^4)*deriv(v[i - 1], x))/(40*n - 40 + (80*n^2 + 48*n - 128)*x + (64*n^3 + 64*n^2 - 128*n)*x^2)); 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

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. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.