Real-rooted atlas

A125662 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12\,n - 8\,n x + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-24 - 52\,x - 32\,x^{2} - 24\,n - 64\,n x - 32\,n x^{2} + 16\,n x^{3} - 12\,n^{2} x + 16\,n^{2} x^{2} + 32\,n^{2} x^{3} + 16\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-9 - 36\,x - 44\,x^{2} - 16\,x^{3} + 12\,n x^{2} + 32\,n x^{3} + 16\,n x^{4} + 12\,n^{2} x^{2} + 32\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2156448127
3563685762561256
4209185644644096128011905
578087282841639936256006144109604
629113929616156830720030080014745628672987903
7108641717448536322050560271360020275208028161310728761808

All 15 rows

Technical details
Node id
n-20111fc1e9216d23 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8ae3b92bdc747eb3

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:      4,       8;
  2:     15,      64,       48;
  3:     56,     368,      576,      256;
  4:    209,    1856,     4464,     4096,      1280;
  5:    780,    8728,    28416,    39936,     25600,      6144;
  6:   2911,   39296,   161568,   307200,    300800,    147456,     28672;
  7:  10864,  171744,   853632,  2050560,   2713600,   2027520,    802816,   131072;
  8:  40545,  734592,  4283424, 12447744,  20806400,  20987904,  12644352,  4194304,   589824;
  9: 151316, 3090984, 20687616, 70585344, 142750720, 181794816, 147718144, 74448896, 21233664, 2621440;
%F Formula
The row polynomials satisfy -(12*n+8*n*x-8*n^3*x^2)*P(n,x) = -(24*n+24+(12*n^2+64*n+52)*x-(16*n^3+16*n^2-32*n-32)*x^2-(16*n^3+32*n^2+16*n)*x^3)*P(n-1,x) - (9+36*x-(12*n^2+12*n-44)*x^2-(32*n^2+32*n-16)*x^3-(16*n^2+16*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[(-(24*n + 24 + (12*n^2 + 64*n + 52)*x - (16*n^3 + 16*n^2 - 32*n - 32)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*P[n - 1, x] - (9 + 36*x - (12*n^2 + 12*n - 44)*x^2 - (32*n^2 + 32*n - 16)*x^3 - (16*n^2 + 16*n)*x^4)*D[P[n - 1, x], x])/(-12*n - 8*n*x + 8*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] = (-(24*n + 24 + (12*n^2 + 64*n + 52)*x - (16*n^3 + 16*n^2 - 32*n - 32)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*v[i - 1] - (9 + 36*x - (12*n^2 + 12*n - 44)*x^2 - (32*n^2 + 32*n - 16)*x^3 - (16*n^2 + 16*n)*x^4)*deriv(v[i - 1], x))/(-12*n - 8*n*x + 8*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