Real-rooted atlas

A104684 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12 + 16\,x + 20\,n - 32\,n x - 11\,n^{2} + 24\,n^{2} x + 2\,n^{3} - 8\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(-36 - 18\,x + 16\,x^{2} + 66\,n + 12\,n x - 46\,n x^{2} - 40\,n^{2} + 18\,n^{2} x + 47\,n^{2} x^{2} + 8\,n^{3} - 18\,n^{3} x - 20\,n^{3} x^{2} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 5\,x^{2} + 4\,x^{3} + 12\,n x + 2\,n x^{2} - 10\,n x^{3} - 4\,n^{2} x + 4\,n^{2} x^{2} + 8\,n^{2} x^{3} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
36410
43048987
514036018016696
663022401890480255265
727721260015120672010503638298
812012665281039506720018900201649270655

All 15 rows

Technical details
Node id
n-3b76d56c9dfaad44 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-980e267510c7a2f6

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:      0;
   2:      1;
   3:      6,       4;
   4:     30,      48,       9;
   5:    140,     360,     180,      16;
   6:    630,    2240,    1890,     480,      25;
   7:   2772,   12600,   15120,    6720,    1050,     36;
   8:  12012,   66528,  103950,   67200,   18900,   2016,    49;
   9:  51480,  336336,  648648,  554400,  231000,  45360,  3528,   64;
  10: 218790, 1647360, 3783780, 4036032, 2252250, 665280, 97020, 5760, 81;
%F Formula
The row polynomials satisfy (2*n^3-11*n^2+20*n-12+(n^4-8*n^3+24*n^2-32*n+16)*x)*P(n,x) = (8*n^3-40*n^2+66*n-36+(4*n^4-18*n^3+18*n^2+12*n-18)*x+(3*n^4-20*n^3+47*n^2-46*n+16)*x^2)*P(n-1,x) - ((4*n^2-12*n+9)*x+(2*n^3-4*n^2-2*n+5)*x^2+(2*n^3-8*n^2+10*n-4)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 40*n^2 + 66*n - 36 + (4*n^4 - 18*n^3 + 18*n^2 + 12*n - 18)*x + (3*n^4 - 20*n^3 + 47*n^2 - 46*n + 16)*x^2)*P[n - 1, x] - ((4*n^2 - 12*n + 9)*x + (2*n^3 - 4*n^2 - 2*n + 5)*x^2 + (2*n^3 - 8*n^2 + 10*n - 4)*x^3)*D[P[n - 1, x], x])/(2*n^3 - 11*n^2 + 20*n - 12 + (n^4 - 8*n^3 + 24*n^2 - 32*n + 16)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^3 - 40*n^2 + 66*n - 36 + (4*n^4 - 18*n^3 + 18*n^2 + 12*n - 18)*x + (3*n^4 - 20*n^3 + 47*n^2 - 46*n + 16)*x^2)*v[i - 1] - ((4*n^2 - 12*n + 9)*x + (2*n^3 - 4*n^2 - 2*n + 5)*x^2 + (2*n^3 - 8*n^2 + 10*n - 4)*x^3)*deriv(v[i - 1], x))/(2*n^3 - 11*n^2 + 20*n - 12 + (n^4 - 8*n^3 + 24*n^2 - 32*n + 16)*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