Real-rooted atlas

A079638 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-48 + 112\,n - 92\,n^{2} + 32\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-36 + 8\,x + 78\,n - 12\,n x - 54\,n^{2} + 4\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-13 + 6\,x - x^{2} + 26\,n - 6\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 9 + 2\,x\)
\(P_{3}(x) = 83 + 36\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
29211
383363122
48705509041514
51047485502055180521264
61438921418484441557403156336216
7223335625483929728671646401337050475933136

All 15 rows

Technical details
Node id
n-47afaee86b7a6b82 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ca763dc598cfee6a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         0;
  1:         1;
  2:         9,          2;
  3:        83,         36,         3;
  4:       870,        550,        90,         4;
  5:     10474,       8550,      2055,       180,        5;
  6:    143892,     141848,     44415,      5740,      315,       6;
  7:   2233356,    2548392,    972867,    164640,    13370,     504,     7;
  8:  38769840,   49741144,  22196916,   4605636,   491400,   27468,   756,    8;
  9: 745785936, 1053582120, 533835600, 130239900, 17099565, 1258740, 51450, 1080, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^4-32*n^3+92*n^2-112*n+48)*P(n-4,x) + (12*n^3-54*n^2+78*n-36+(4*n^2-12*n+8)*x)*P(n-3,x) - (13*n^2-26*n+13+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 9 + 2*x, P(3,x) = 83 + 36*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 9 + 2*x;
P[3, x_] := 83 + 36*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*P[n - 4, x] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*P[n - 3, x] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 9 + 2*x; v[4] = 83 + 36*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*v[i - 4] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*v[i - 3] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*v[i - 2] + (6*n - 3 + 2*x)*v[i - 1]); 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