Real-rooted atlas

A391045 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 4\,n^{2} - 2\,n^{2} x + 8\,n^{3} + 2\,n^{3} x + 6\,n^{4} x - 10\,n^{5} x + 4\,n^{6} x\right)\,P_{n}(x) = \left(-4\,n + 16\,n x + 4\,n x^{2} - 4\,n^{2} - 46\,n^{2} x - 14\,n^{2} x^{2} + 8\,n^{3} + 30\,n^{3} x + 14\,n^{3} x^{2} + 6\,n^{4} x + 5\,n^{4} x^{2} - 10\,n^{5} x - 16\,n^{5} x^{2} + 4\,n^{6} x + 7\,n^{6} x^{2}\right)\,P_{n-1}(x) + \left(-24\,x + 12\,x^{2} + 6\,x^{3} + 36\,n x - 34\,n x^{2} - 18\,n x^{3} + 24\,n^{2} x + 20\,n^{2} x^{2} + 12\,n^{2} x^{3} - 22\,n^{3} x^{2} + 14\,n^{3} x^{3} - 6\,n^{4} x^{2} - 16\,n^{4} x^{3} + 12\,n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}'(x) + \left(-16\,x^{2} - 4\,x^{3} + 2\,x^{4} + 40\,n x^{2} + 14\,n x^{3} - 6\,n x^{4} - 16\,n^{2} x^{3} + 4\,n^{2} x^{4} - 20\,n^{3} x^{3} + 5\,n^{3} x^{4} + 20\,n^{4} x^{3} - 5\,n^{4} x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
318312
412027452
5140126645236
617042048012561102
7111211342400137521675245
811682646924096253276343825307

All 15 rows

Technical details
Node id
n-6a3feb59f46c915f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-cbb0d2a2a7c53ac6

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   2;
   3: 1,   8,     3;
   4: 1,  20,    27,     4;
   5: 1,  40,   126,    64,      5;
   6: 1,  70,   420,   480,    125,      6;
   7: 1, 112,  1134,  2400,   1375,    216,     7;
   8: 1, 168,  2646,  9240,   9625,   3276,   343,     8;
   9: 1, 240,  5544, 29568,  50050,  30576,  6860,   512,   9;
  10: 1, 330, 10692, 82368, 210210, 206388, 82320, 13056, 729, 10;
%F Formula
The row polynomials satisfy (8*n^3-4*n^2-4*n+(4*n^6-10*n^5+6*n^4+2*n^3-2*n^2)*x)*P(n,x) = (8*n^3-4*n^2-4*n+(4*n^6-10*n^5+6*n^4+30*n^3-46*n^2+16*n)*x+(7*n^6-16*n^5+5*n^4+14*n^3-14*n^2+4*n)*x^2)*P(n-1,x) + ((24*n^2+36*n-24)*x+(12*n^5-6*n^4-22*n^3+20*n^2-34*n+12)*x^2+(2*n^5-16*n^4+14*n^3+12*n^2-18*n+6)*x^3)*d/dx P(n-1,x) + ((40*n-16)*x^2+(20*n^4-20*n^3-16*n^2+14*n-4)*x^3-(5*n^4-5*n^3-4*n^2+6*n-2)*x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 30*n^3 - 46*n^2 + 16*n)*x + (7*n^6 - 16*n^5 + 5*n^4 + 14*n^3 - 14*n^2 + 4*n)*x^2)*P[n - 1, x] + ((24*n^2 + 36*n - 24)*x + (12*n^5 - 6*n^4 - 22*n^3 + 20*n^2 - 34*n + 12)*x^2 + (2*n^5 - 16*n^4 + 14*n^3 + 12*n^2 - 18*n + 6)*x^3)*D[P[n - 1, x], x] + ((40*n - 16)*x^2 + (20*n^4 - 20*n^3 - 16*n^2 + 14*n - 4)*x^3 - (5*n^4 - 5*n^3 - 4*n^2 + 6*n - 2)*x^4)*D[P[n - 1, x], {x, 2}])/(8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 2*n^3 - 2*n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 30*n^3 - 46*n^2 + 16*n)*x + (7*n^6 - 16*n^5 + 5*n^4 + 14*n^3 - 14*n^2 + 4*n)*x^2)*v[i - 1] + ((24*n^2 + 36*n - 24)*x + (12*n^5 - 6*n^4 - 22*n^3 + 20*n^2 - 34*n + 12)*x^2 + (2*n^5 - 16*n^4 + 14*n^3 + 12*n^2 - 18*n + 6)*x^3)*deriv(v[i - 1], x) + ((40*n - 16)*x^2 + (20*n^4 - 20*n^3 - 16*n^2 + 14*n - 4)*x^3 - (5*n^4 - 5*n^3 - 4*n^2 + 6*n - 2)*x^4)*deriv(deriv(v[i - 1], x), x))/(8*n^3 - 4*n^2 - 4*n + (4*n^6 - 10*n^5 + 6*n^4 + 2*n^3 - 2*n^2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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