Real-rooted atlas

A078812 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 4\,x - 6\,n + 16\,n x + 4\,n^{2} - 20\,n^{2} x + 8\,n^{3} x\right)\,P_{n}(x) = \left(2 + 12\,x + 16\,x^{2} - 6\,n - 28\,n x - 8\,n x^{2} + 4\,n^{2} + 4\,n^{2} x - 36\,n^{2} x^{2} + 8\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 8\,x^{3} - 2\,n x - 8\,n x^{2} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
318918
4112301659
5116638025185
612010822417536564
7124165480630336491685
812823488016501512588644957

All 15 rows

Technical details
Node id
n-bf7b34a29f09cf83 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c7722c8666024957

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,  4;
   3: 1,  8,   9;
   4: 1, 12,  30,   16;
   5: 1, 16,  63,   80,   25;
   6: 1, 20, 108,  224,  175,    36;
   7: 1, 24, 165,  480,  630,   336,    49;
   8: 1, 28, 234,  880, 1650,  1512,   588,   64;
   9: 1, 32, 315, 1456, 3575,  4752,  3234,  960,   81;
  10: 1, 36, 408, 2240, 6825, 12012, 12012, 6336, 1485, 100;
%F Formula
The row polynomials satisfy (4*n^2-6*n+2+(8*n^3-20*n^2+16*n-4)*x)*P(n,x) = (4*n^2-6*n+2+(8*n^3+4*n^2-28*n+12)*x+(24*n^3-36*n^2-8*n+16)*x^2)*P(n-1,x) - ((2*n-1)*x+(4*n^2+8*n-6)*x^2+(16*n^2-8)*x^3)*d/dx 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[((4*n^2 - 6*n + 2 + (8*n^3 + 4*n^2 - 28*n + 12)*x + (24*n^3 - 36*n^2 - 8*n + 16)*x^2)*P[n - 1, x] - ((2*n - 1)*x + (4*n^2 + 8*n - 6)*x^2 + (16*n^2 - 8)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (8*n^3 - 20*n^2 + 16*n - 4)*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] = ((4*n^2 - 6*n + 2 + (8*n^3 + 4*n^2 - 28*n + 12)*x + (24*n^3 - 36*n^2 - 8*n + 16)*x^2)*v[i - 1] - ((2*n - 1)*x + (4*n^2 + 8*n - 6)*x^2 + (16*n^2 - 8)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 2 + (8*n^3 - 20*n^2 + 16*n - 4)*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

Same fingerprint ?

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