Real-rooted atlas

A033282 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-8 - 4\,n + 2\,n x^{2} + 3\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-24 - 24\,x + 4\,x^{3} - 12\,n - 20\,n x + 6\,n x^{2} + 10\,n x^{3} - 4\,n^{2} x + 9\,n^{2} x^{2} + 8\,n^{2} x^{3} + 3\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 12\,x + 6\,x^{3} + 2\,x^{4} + 8\,n x^{2} + 12\,n x^{3} + 4\,n x^{4} + 4\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
211301556
34518618956476
41971092168010082104187
590362801296011760495079237645
64279358029274511286071775231663003343630
720793203350634095968660825825402402105105114403171670

All 15 rows

Technical details
Node id
n-ff5afdbf2a55582c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8e8ddef9357b0015

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,       4;
  2:     11,      30,       15;
  3:     45,     186,      189,       56;
  4:    197,    1092,     1680,     1008,      210;
  5:    903,    6280,    12960,    11760,     4950,      792;
  6:   4279,   35802,    92745,   112860,    71775,    23166,     3003;
  7:  20793,  203350,   634095,   968660,   825825,   402402,   105105,    11440;
  8: 103049, 1153416,  4206048,  7735728,  8278270,  5405400,  2130128,   466752,   43758;
  9: 518859, 6540876, 27310608, 58753968, 75700170, 62034336, 32746896, 10820160, 2040714, 167960;
%F Formula
The row polynomials satisfy -(4*n+8-(n^3+3*n^2+2*n)*x^2)*P(n,x) = -(12*n+24+(4*n^2+20*n+24)*x-(3*n^3+9*n^2+6*n)*x^2-(2*n^3+8*n^2+10*n+4)*x^3)*P(n-1,x) - (8+12*x-(4*n^2+8*n)*x^2-(6*n^2+12*n+6)*x^3-(2*n^2+4*n+2)*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[(-(12*n + 24 + (4*n^2 + 20*n + 24)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (2*n^3 + 8*n^2 + 10*n + 4)*x^3)*P[n - 1, x] - (8 + 12*x - (4*n^2 + 8*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (2*n^2 + 4*n + 2)*x^4)*D[P[n - 1, x], x])/(-4*n - 8 + (n^3 + 3*n^2 + 2*n)*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] = (-(12*n + 24 + (4*n^2 + 20*n + 24)*x - (3*n^3 + 9*n^2 + 6*n)*x^2 - (2*n^3 + 8*n^2 + 10*n + 4)*x^3)*v[i - 1] - (8 + 12*x - (4*n^2 + 8*n)*x^2 - (6*n^2 + 12*n + 6)*x^3 - (2*n^2 + 4*n + 2)*x^4)*deriv(v[i - 1], x))/(-4*n - 8 + (n^3 + 3*n^2 + 2*n)*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