Real-rooted atlas

A207607 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(200 + 1600\,x + 3456\,x^{2} - 80\,n - 1216\,n x - 2624\,n x^{2} + 2048\,n x^{3} - 960\,n^{2} x^{2} - 3072\,n^{2} x^{3} + 128\,n^{3} x^{2} + 1024\,n^{3} x^{3}\right)\,P_{n}(x) = \left(480 + 4896\,x + 15360\,x^{2} + 14336\,x^{3} + 4096\,x^{4} - 120\,n - 1744\,n x - 7040\,n x^{2} - 8704\,n x^{3} - 2048\,n x^{4} - 80\,n^{2} x - 2368\,n^{2} x^{2} - 7424\,n^{2} x^{3} - 4096\,n^{2} x^{4} + 192\,n^{3} x^{2} + 1792\,n^{3} x^{3} + 2048\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-25 - 680\,x - 5328\,x^{2} - 15360\,x^{3} - 14336\,x^{4} - 4096\,x^{5} - 320\,n x^{2} - 2176\,n x^{3} - 4096\,n x^{4} - 2048\,n x^{5} + 80\,n^{2} x^{2} + 1024\,n^{2} x^{3} + 3328\,n^{2} x^{4} + 2048\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131619
287296176
3212647205121517
45588836485376256012527
5144284015552350723456012288100456
637787846004818278427648020275257344788569
7987265282172968335361708800192307211182082621446090571

All 15 rows

Technical details
Node id
n-f0fa8298e4144299 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a91d34df3a622d43

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:    3,     16;
  2:    8,     72,      96;
  3:   21,    264,     720,      512;
  4:   55,    888,    3648,     5376,     2560;
  5:  144,   2840,   15552,    35072,    34560,    12288;
  6:  377,   8784,   60048,   182784,   276480,   202752,    57344;
  7:  987,  26528,  217296,   833536,  1708800,  1923072,  1118208,   262144;
  8: 2584,  78696,  751008,  3476736,  9017600, 13768704, 12271616,  5898240,  1179648;
  9: 6765, 230232, 2507904, 13602048, 42727680, 82667520, 99950592, 73531392, 30081024, 5242880;
%F Formula
The row polynomials satisfy -(80*n-200+(1216*n-1600)*x-(128*n^3-960*n^2-2624*n+3456)*x^2-(1024*n^3-3072*n^2+2048*n)*x^3)*P(n,x) = -(120*n-480+(80*n^2+1744*n-4896)*x-(192*n^3-2368*n^2-7040*n+15360)*x^2-(1792*n^3-7424*n^2-8704*n+14336)*x^3-(2048*n^3-4096*n^2-2048*n+4096)*x^4)*P(n-1,x) - (25+680*x-(80*n^2-320*n-5328)*x^2-(1024*n^2-2176*n-15360)*x^3-(3328*n^2-4096*n-14336)*x^4-(2048*n^2-2048*n-4096)*x^5)*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[(-(120*n - 480 + (80*n^2 + 1744*n - 4896)*x - (192*n^3 - 2368*n^2 - 7040*n + 15360)*x^2 - (1792*n^3 - 7424*n^2 - 8704*n + 14336)*x^3 - (2048*n^3 - 4096*n^2 - 2048*n + 4096)*x^4)*P[n - 1, x] - (25 + 680*x - (80*n^2 - 320*n - 5328)*x^2 - (1024*n^2 - 2176*n - 15360)*x^3 - (3328*n^2 - 4096*n - 14336)*x^4 - (2048*n^2 - 2048*n - 4096)*x^5)*D[P[n - 1, x], x])/(-80*n + 200 - (1216*n - 1600)*x + (128*n^3 - 960*n^2 - 2624*n + 3456)*x^2 + (1024*n^3 - 3072*n^2 + 2048*n)*x^3)]];
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] = (-(120*n - 480 + (80*n^2 + 1744*n - 4896)*x - (192*n^3 - 2368*n^2 - 7040*n + 15360)*x^2 - (1792*n^3 - 7424*n^2 - 8704*n + 14336)*x^3 - (2048*n^3 - 4096*n^2 - 2048*n + 4096)*x^4)*v[i - 1] - (25 + 680*x - (80*n^2 - 320*n - 5328)*x^2 - (1024*n^2 - 2176*n - 15360)*x^3 - (3328*n^2 - 4096*n - 14336)*x^4 - (2048*n^2 - 2048*n - 4096)*x^5)*deriv(v[i - 1], x))/(-80*n + 200 - (1216*n - 1600)*x + (128*n^3 - 960*n^2 - 2624*n + 3456)*x^2 + (1024*n^3 - 3072*n^2 + 2048*n)*x^3)); 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. [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