Real-rooted atlas

A145597 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(36\,n - 66\,n x + 24\,n x^{2} - 12\,n^{2} + 13\,n^{2} x + 4\,n^{2} x^{2} + 3\,n^{3} + 4\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(48 - 156\,x + 104\,x^{2} - 32\,x^{3} + 20\,n - 4\,n x - 28\,n x^{2} + 16\,n x^{3} + 8\,n^{2} + 13\,n^{2} x - 8\,n^{2} x^{2} + 16\,n^{2} x^{3} + 7\,n^{3} - 4\,n^{3} x + 16\,n^{3} x^{2} + n^{4} + 4\,n^{4} x\right)\,P_{n-1}(x) + \left(12\,x - 60\,x^{2} + 64\,x^{3} - 64\,x^{4} + 2\,n x + 2\,n x^{2} - 32\,n x^{3} - 32\,n x^{4} + 2\,n^{2} x - 32\,n^{2} x^{3} + 2\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
29312
3251540
460639132
512623163420
6238714336281316
7414189015122524068

All 15 rows

Technical details
Node id
n-23435bf031ad265a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-965dd705f7c88718

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:    9,    3;
  3:   25,   15;
  4:   60,   63,     9;
  5:  126,  231,    63;
  6:  238,  714,   336,   28;
  7:  414, 1890,  1512,  252;
  8:  675, 4410,  5670, 1620,  90;
  9: 1045, 9306, 18018, 8580, 990;
%F Formula
The row polynomials satisfy (n^4+3*n^3-12*n^2+36*n+(4*n^3+13*n^2-66*n)*x+(4*n^2+24*n)*x^2)*P(n,x) = (n^4+7*n^3+8*n^2+20*n+48+(4*n^4-4*n^3+13*n^2-4*n-156)*x+(16*n^3-8*n^2-28*n+104)*x^2+(16*n^2+16*n-32)*x^3)*P(n-1,x) + ((2*n^3+2*n^2+2*n+12)*x-(8*n^3-2*n+60)*x^2-(32*n^2+32*n-64)*x^3-(32*n+64)*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[((n^4 + 7*n^3 + 8*n^2 + 20*n + 48 + (4*n^4 - 4*n^3 + 13*n^2 - 4*n - 156)*x + (16*n^3 - 8*n^2 - 28*n + 104)*x^2 + (16*n^2 + 16*n - 32)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2 + 2*n + 12)*x - (8*n^3 - 2*n + 60)*x^2 - (32*n^2 + 32*n - 64)*x^3 - (32*n + 64)*x^4)*D[P[n - 1, x], x])/(n^4 + 3*n^3 - 12*n^2 + 36*n + (4*n^3 + 13*n^2 - 66*n)*x + (4*n^2 + 24*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] = ((n^4 + 7*n^3 + 8*n^2 + 20*n + 48 + (4*n^4 - 4*n^3 + 13*n^2 - 4*n - 156)*x + (16*n^3 - 8*n^2 - 28*n + 104)*x^2 + (16*n^2 + 16*n - 32)*x^3)*v[i - 1] + ((2*n^3 + 2*n^2 + 2*n + 12)*x - (8*n^3 - 2*n + 60)*x^2 - (32*n^2 + 32*n - 64)*x^3 - (32*n + 64)*x^4)*deriv(v[i - 1], x))/(n^4 + 3*n^3 - 12*n^2 + 36*n + (4*n^3 + 13*n^2 - 66*n)*x + (4*n^2 + 24*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