Real-rooted atlas

A232535 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-19\,x + 6\,x^{2} - 9\,n + 30\,n x - 19\,n x^{2} + 6\,n x^{3} + 15\,n^{2} - 17\,n^{2} x + 12\,n^{2} x^{2} - 6\,n^{3} + 6\,n^{3} x\right)\,P_{n}(x) = \left(-36 - 144\,x - 74\,x^{2} + 69\,x^{3} - 18\,x^{4} + 24\,n + 125\,n x + 98\,n x^{2} - 129\,n x^{3} + 54\,n x^{4} + 36\,n^{2} + 49\,n^{2} x - 87\,n^{2} x^{2} + 108\,n^{2} x^{3} - 24\,n^{3} - 30\,n^{3} x + 54\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x - 46\,x^{2} - 16\,x^{3} + 30\,x^{4} - 36\,x^{5} + 16\,n x^{2} + 24\,n x^{3} - 72\,n x^{4} + 12\,n^{2} x + 24\,n^{2} x^{2} - 36\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
2549627177
32889066571081959
4144065768820367240520913
56912415688380865772182251458217743
6322562411526554887909924168268359251032225409
7147456131942445319687502976617414423643903623131749622420167

All 15 rows

Technical details
Node id
n-45c08373a943b8b4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8a8ae75bcd9de4e8

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 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:       9,        6;
  2:      54,       96,        27;
  3:     288,      906,       657,       108;
  4:    1440,     6576,      8820,      3672,       405;
  5:    6912,    41568,     83808,     65772,     18225,      1458;
  6:   32256,   241152,    655488,    790992,    416826,     83592,      5103;
  7:  147456,  1319424,   4531968,   7502976,   6174144,   2364390,    362313,    17496;
  8:  663552,  6918144,  28781568,  61005312,  69802560,  42200352,  12369672,  1504656,   59049;
  9: 2949120, 35119104, 171786240, 445436928, 659508480, 560260800, 261320256, 60833592, 6042681, 196830;
%F Formula
The row polynomials satisfy -(6*n^3-15*n^2+9*n-(6*n^3-17*n^2+30*n-19)*x-(12*n^2-19*n+6)*x^2-6*n*x^3)*P(n,x) = -(24*n^3-36*n^2-24*n+36+(30*n^3-49*n^2-125*n+144)*x-(54*n^3-87*n^2+98*n-74)*x^2-(108*n^2-129*n+69)*x^3-(54*n-18)*x^4)*P(n-1,x) + ((12*n^2-12)*x+(24*n^2+16*n-46)*x^2-(36*n^2-24*n+16)*x^3-(72*n-30)*x^4-36*x^5)*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[(-(24*n^3 - 36*n^2 - 24*n + 36 + (30*n^3 - 49*n^2 - 125*n + 144)*x - (54*n^3 - 87*n^2 + 98*n - 74)*x^2 - (108*n^2 - 129*n + 69)*x^3 - (54*n - 18)*x^4)*P[n - 1, x] + ((12*n^2 - 12)*x + (24*n^2 + 16*n - 46)*x^2 - (36*n^2 - 24*n + 16)*x^3 - (72*n - 30)*x^4 - 36*x^5)*D[P[n - 1, x], x])/(-6*n^3 + 15*n^2 - 9*n + (6*n^3 - 17*n^2 + 30*n - 19)*x + (12*n^2 - 19*n + 6)*x^2 + 6*n*x^3)]];
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] = (-(24*n^3 - 36*n^2 - 24*n + 36 + (30*n^3 - 49*n^2 - 125*n + 144)*x - (54*n^3 - 87*n^2 + 98*n - 74)*x^2 - (108*n^2 - 129*n + 69)*x^3 - (54*n - 18)*x^4)*v[i - 1] + ((12*n^2 - 12)*x + (24*n^2 + 16*n - 46)*x^2 - (36*n^2 - 24*n + 16)*x^3 - (72*n - 30)*x^4 - 36*x^5)*deriv(v[i - 1], x))/(-6*n^3 + 15*n^2 - 9*n + (6*n^3 - 17*n^2 + 30*n - 19)*x + (12*n^2 - 19*n + 6)*x^2 + 6*n*x^3)); 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. [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