Real-rooted atlas

A207824 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-8\,n - 3\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-24 - 33\,x - 12\,x^{2} - 24\,n - 41\,n x - 12\,n x^{2} + 4\,n x^{3} - 8\,n^{2} x + 6\,n^{2} x^{2} + 8\,n^{2} x^{3} + 6\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-16 - 36\,x - 26\,x^{2} - 6\,x^{3} + 8\,n x^{2} + 12\,n x^{3} + 4\,n x^{4} + 8\,n^{2} x^{2} + 12\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16814
2359648179
32048488642562172
411896624102246144128025461
569304840810022491136384006144291242
6403913392648815681075200684800221184286723271079
7235416231030472247681106176094464004608000120422413107236221944

All 15 rows

Technical details
Node id
n-9ab828603d7e9388 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-aa60f51a0b464238

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:       6,         8;
  2:      35,        96,        48;
  3:     204,       848,       864,       256;
  4:    1189,      6624,     10224,      6144,       1280;
  5:    6930,     48408,    100224,     91136,      38400,      6144;
  6:   40391,    339264,    881568,   1075200,     684800,    221184,     28672;
  7:  235416,   2310304,   7224768,  11061760,    9446400,   4608000,   1204224,    131072;
  8: 1372105,  15405888,  56328864, 103827456,  111302400,  72769536,  28700672,   6291456,   589824;
  9: 7997214, 101101864, 423183744, 912323584, 1177505280, 966260736, 510590976, 168820736, 31850496, 2621440;
%F Formula
The row polynomials satisfy -(8*n+3*n*x-2*n^3*x^2)*P(n,x) = -(24*n+24+(8*n^2+41*n+33)*x-(6*n^3+6*n^2-12*n-12)*x^2-(4*n^3+8*n^2+4*n)*x^3)*P(n-1,x) - (16+36*x-(8*n^2+8*n-26)*x^2-(12*n^2+12*n-6)*x^3-(4*n^2+4*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 + (8*n^2 + 41*n + 33)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*P[n - 1, x] - (16 + 36*x - (8*n^2 + 8*n - 26)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (4*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(-8*n - 3*n*x + 2*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 + (8*n^2 + 41*n + 33)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*v[i - 1] - (16 + 36*x - (8*n^2 + 8*n - 26)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (4*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(-8*n - 3*n*x + 2*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