Real-rooted atlas

A078812 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-18 + 34\,x - 8\,x^{2} + 42\,n - 50\,n x + 8\,n x^{2} - 32\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(-6 + 2\,x + 96\,x^{2} - 32\,x^{3} + 22\,n + 46\,n x - 134\,n x^{2} + 24\,n x^{3} - 24\,n^{2} - 72\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x + x^{2} + 32\,x^{3} - 16\,x^{4} + 6\,n x + 16\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
3510318
472622459
595081405185
61182204196656564
7131224156244069871685
8151707381540159075614084957

All 15 rows

Technical details
Node id
n-5bea5b93ff0462bc · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2aa7c5ebcb2842d2

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:  3,   2;
   3:  5,  10,    3;
   4:  7,  26,   22,    4;
   5:  9,  50,   81,   40,     5;
   6: 11,  82,  204,  196,    65,     6;
   7: 13, 122,  415,  624,   406,    98,    7;
   8: 15, 170,  738, 1540,  1590,   756,  140,    8;
   9: 17, 226, 1197, 3224,  4675,  3564, 1302,  192,   9;
  10: 19, 290, 1816, 6020, 11375, 12298, 7260, 2112, 255, 10;
%F Formula
The row polynomials satisfy (8*n^3-32*n^2+42*n-18+(16*n^2-50*n+34)*x+(8*n-8)*x^2)*P(n,x) = (8*n^3-24*n^2+22*n-6+(24*n^3-72*n^2+46*n+2)*x+(48*n^2-134*n+96)*x^2+(24*n-32)*x^3)*P(n-1,x) - ((4*n^2-6*n+2)*x+(16*n^2-16*n-1)*x^2+(32*n-32)*x^3+16*x^4)*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[((8*n^3 - 24*n^2 + 22*n - 6 + (24*n^3 - 72*n^2 + 46*n + 2)*x + (48*n^2 - 134*n + 96)*x^2 + (24*n - 32)*x^3)*P[n - 1, x] - ((4*n^2 - 6*n + 2)*x + (16*n^2 - 16*n - 1)*x^2 + (32*n - 32)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 32*n^2 + 42*n - 18 + (16*n^2 - 50*n + 34)*x + (8*n - 8)*x^2)]];
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] = ((8*n^3 - 24*n^2 + 22*n - 6 + (24*n^3 - 72*n^2 + 46*n + 2)*x + (48*n^2 - 134*n + 96)*x^2 + (24*n - 32)*x^3)*v[i - 1] - ((4*n^2 - 6*n + 2)*x + (16*n^2 - 16*n - 1)*x^2 + (32*n - 32)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 32*n^2 + 42*n - 18 + (16*n^2 - 50*n + 34)*x + (8*n - 8)*x^2)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023