Real-rooted atlas

A069466 → γ → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-n^{2} + n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(-16\,x + 16\,x^{2} + 2\,n + 48\,n x - 24\,n x^{2} - 4\,n^{2} - 34\,n^{2} x - 32\,n^{2} x^{2} + 2\,n^{3} x + 24\,n^{3} x^{2} + 4\,n^{4} x + 16\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(4\,x - 24\,x^{2} + 32\,x^{3} - 8\,n x + 36\,n x^{2} - 16\,n x^{3} + 20\,n^{2} x^{2} - 80\,n^{2} x^{3} + 8\,n^{3} x^{2} - 32\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
262430
320240260
470168012603010
5252100802268033012
69245544024948073920379764
73432288288216216019219204375800

All 15 rows

Technical details
Node id
n-e182fb58f6f67370 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-adc8add1ebe9f8d3

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:     2;
  2:     6,      24;
  3:    20,     240;
  4:    70,    1680,      1260;
  5:   252,   10080,     22680;
  6:   924,   55440,    249480,     73920;
  7:  3432,  288288,   2162160,   1921920;
  8: 12870, 1441440,  16216200,  28828800,   4504500;
  9: 48620, 7001280, 110270160, 326726400, 153153000;
%F Formula
The row polynomials satisfy -(n^2-(n^4+n^3)*x)*P(n,x) = -(4*n^2-2*n-(4*n^4+2*n^3-34*n^2+48*n-16)*x-(16*n^4+24*n^3-32*n^2-24*n+16)*x^2)*P(n-1,x) - ((8*n-4)*x-(8*n^3+20*n^2+36*n-24)*x^2+(32*n^3+80*n^2+16*n-32)*x^3)*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[(-(4*n^2 - 2*n - (4*n^4 + 2*n^3 - 34*n^2 + 48*n - 16)*x - (16*n^4 + 24*n^3 - 32*n^2 - 24*n + 16)*x^2)*P[n - 1, x] - ((8*n - 4)*x - (8*n^3 + 20*n^2 + 36*n - 24)*x^2 + (32*n^3 + 80*n^2 + 16*n - 32)*x^3)*D[P[n - 1, x], x])/(-n^2 + (n^4 + n^3)*x)]];
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] = (-(4*n^2 - 2*n - (4*n^4 + 2*n^3 - 34*n^2 + 48*n - 16)*x - (16*n^4 + 24*n^3 - 32*n^2 - 24*n + 16)*x^2)*v[i - 1] - ((8*n - 4)*x - (8*n^3 + 20*n^2 + 36*n - 24)*x^2 + (32*n^3 + 80*n^2 + 16*n - 32)*x^3)*deriv(v[i - 1], x))/(-n^2 + (n^4 + n^3)*x)); 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

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