Real-rooted atlas

A116414 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-208\,n - 160\,n x + 128\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-520 - 1024\,x - 640\,x^{2} - 520\,n - 1232\,n x - 640\,n x^{2} + 256\,n x^{3} - 208\,n^{2} x + 320\,n^{2} x^{2} + 512\,n^{2} x^{3} + 320\,n^{3} x^{2} + 256\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 780\,x - 1008\,x^{2} - 320\,x^{3} + 208\,n x^{2} + 640\,n x^{3} + 256\,n x^{4} + 208\,n^{2} x^{2} + 640\,n^{2} x^{3} + 256\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15813
2228048150
3955527202561623
4409328067685120128016857
51760180165136060928320006144170208
6757394320344592563200460800184320286721683477
7325854781362134800447104050240003115008100352013107216390161

All 15 rows

Technical details
Node id
n-fd107ab37ea858c8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b6eb163d6aec8863

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:      5,        8;
  2:     22,       80,       48;
  3:     95,      552,      720,       256;
  4:    409,     3280,     6768,      5120,      1280;
  5:   1760,    18016,    51360,     60928,     32000,      6144;
  6:   7573,    94320,   344592,    563200,    460800,    184320,     28672;
  7:  32585,   478136,  2134800,   4471040,   5024000,   3115008,   1003520,    131072;
  8: 140206,  2368400, 12509040,  32051200,  46092800,  39137280,  19468288,   5242880,   589824;
  9: 603275, 11529240, 70351200, 213557760, 375648000, 407586816, 276971520, 114819072, 26542080, 2621440;
%F Formula
The row polynomials satisfy -(208*n+160*n*x-128*n^3*x^2)*P(n,x) = -(520*n+520+(208*n^2+1232*n+1024)*x-(320*n^3+320*n^2-640*n-640)*x^2-(256*n^3+512*n^2+256*n)*x^3)*P(n-1,x) - (169+780*x-(208*n^2+208*n-1008)*x^2-(640*n^2+640*n-320)*x^3-(256*n^2+256*n)*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[(-(520*n + 520 + (208*n^2 + 1232*n + 1024)*x - (320*n^3 + 320*n^2 - 640*n - 640)*x^2 - (256*n^3 + 512*n^2 + 256*n)*x^3)*P[n - 1, x] - (169 + 780*x - (208*n^2 + 208*n - 1008)*x^2 - (640*n^2 + 640*n - 320)*x^3 - (256*n^2 + 256*n)*x^4)*D[P[n - 1, x], x])/(-208*n - 160*n*x + 128*n^3*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] = (-(520*n + 520 + (208*n^2 + 1232*n + 1024)*x - (320*n^3 + 320*n^2 - 640*n - 640)*x^2 - (256*n^3 + 512*n^2 + 256*n)*x^3)*v[i - 1] - (169 + 780*x - (208*n^2 + 208*n - 1008)*x^2 - (640*n^2 + 640*n - 320)*x^3 - (256*n^2 + 256*n)*x^4)*deriv(v[i - 1], x))/(-208*n - 160*n*x + 128*n^3*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. [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