Real-rooted atlas

A069466 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n^{2} - 11\,n^{2} x + 4\,n^{2} x^{2} - 3\,n^{3} + 4\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(-56\,x^{2} + 32\,x^{3} - 2\,n - 2\,n x + 192\,n x^{2} - 96\,n x^{3} + 6\,n^{2} + 28\,n^{2} x - 192\,n^{2} x^{2} + 64\,n^{2} x^{3} - 6\,n^{3} - 56\,n^{3} x + 64\,n^{3} x^{2} + 4\,n^{4} + 16\,n^{4} x\right)\,P_{n-1}(x) + \left(12\,x^{2} - 64\,x^{3} + 64\,x^{4} + 4\,n x - 56\,n x^{2} + 192\,n x^{3} - 128\,n x^{4} - 12\,n^{2} x + 80\,n^{2} x^{2} - 128\,n^{2} x^{3} + 8\,n^{3} x - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
2181230
3140120260
491016804203010
5529220160756033012
62864419404013860018480379764
7147576158558421621604804804375800

All 15 rows

Technical details
Node id
n-ed70e8d718aba50c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4806a24fb0d130c1

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:      18,       12;
  3:     140,      120;
  4:     910,     1680,       420;
  5:    5292,    20160,      7560;
  6:   28644,   194040,    138600,     18480;
  7:  147576,  1585584,   2162160,    480480;
  8:  733590, 11531520,  27027000,  10810800,   900900;
  9: 3549260, 77014080, 281801520, 204204000, 30630600;
%F Formula
The row polynomials satisfy (n^4-3*n^3+3*n^2+(4*n^3-11*n^2)*x+4*n^2*x^2)*P(n,x) = (4*n^4-6*n^3+6*n^2-2*n+(16*n^4-56*n^3+28*n^2-2*n)*x+(64*n^3-192*n^2+192*n-56)*x^2+(64*n^2-96*n+32)*x^3)*P(n-1,x) + ((8*n^3-12*n^2+4*n)*x-(32*n^3-80*n^2+56*n-12)*x^2-(128*n^2-192*n+64)*x^3-(128*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[((4*n^4 - 6*n^3 + 6*n^2 - 2*n + (16*n^4 - 56*n^3 + 28*n^2 - 2*n)*x + (64*n^3 - 192*n^2 + 192*n - 56)*x^2 + (64*n^2 - 96*n + 32)*x^3)*P[n - 1, x] + ((8*n^3 - 12*n^2 + 4*n)*x - (32*n^3 - 80*n^2 + 56*n - 12)*x^2 - (128*n^2 - 192*n + 64)*x^3 - (128*n - 64)*x^4)*D[P[n - 1, x], x])/(n^4 - 3*n^3 + 3*n^2 + (4*n^3 - 11*n^2)*x + 4*n^2*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] = ((4*n^4 - 6*n^3 + 6*n^2 - 2*n + (16*n^4 - 56*n^3 + 28*n^2 - 2*n)*x + (64*n^3 - 192*n^2 + 192*n - 56)*x^2 + (64*n^2 - 96*n + 32)*x^3)*v[i - 1] + ((8*n^3 - 12*n^2 + 4*n)*x - (32*n^3 - 80*n^2 + 56*n - 12)*x^2 - (128*n^2 - 192*n + 64)*x^3 - (128*n - 64)*x^4)*deriv(v[i - 1], x))/(n^4 - 3*n^3 + 3*n^2 + (4*n^3 - 11*n^2)*x + 4*n^2*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

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