Real-rooted atlas

A091044 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(n - 9\,n x + 4\,n x^{2} - 4\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(20\,x^{2} - 16\,x^{3} - 4\,n - 52\,n x - 76\,n x^{2} + 48\,n x^{3} + 96\,n^{2} x^{2} + 16\,n^{3} + 48\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - 32\,x^{4} - 4\,n x - 32\,n x^{2} - 64\,n x^{3} - 8\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112820
28016048288
3448166414082563776
42304128002073610240128047360
51126483968208896200704665606144577536
653248499712169984025559041662976401408286726901760
7245760278528012091392252313602605056012386304229376013107281215488

All 15 rows

Technical details
Node id
n-360a41dc79d33957 · 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 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:      12,        8;
  2:      80,      160,        48;
  3:     448,     1664,      1408,        256;
  4:    2304,    12800,     20736,      10240,       1280;
  5:   11264,    83968,    208896,     200704,      66560,       6144;
  6:   53248,   499712,   1699840,    2555904,    1662976,     401408,      28672;
  7:  245760,  2785280,  12091392,   25231360,   26050560,   12386304,    2293760,    131072;
  8: 1114112, 14811136,  78446592,  211288064,  306380800,  233570304,   85327872,  12582912,   589824;
  9: 4980736, 76021760, 476053504, 1578106880, 2981888000, 3223846912, 1903165440, 553648128, 66846720, 2621440;
%F Formula
The row polynomials satisfy (4*n^3-4*n^2+n+(8*n^2-9*n)*x+4*n*x^2)*P(n,x) = (16*n^3-4*n+(48*n^3-52*n)*x+(96*n^2-76*n+20)*x^2+(48*n-16)*x^3)*P(n-1,x) - ((8*n^2+4*n)*x+(32*n^2+32*n-2)*x^2+64*n*x^3+32*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[((16*n^3 - 4*n + (48*n^3 - 52*n)*x + (96*n^2 - 76*n + 20)*x^2 + (48*n - 16)*x^3)*P[n - 1, x] - ((8*n^2 + 4*n)*x + (32*n^2 + 32*n - 2)*x^2 + 64*n*x^3 + 32*x^4)*D[P[n - 1, x], x])/(4*n^3 - 4*n^2 + n + (8*n^2 - 9*n)*x + 4*n*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] = ((16*n^3 - 4*n + (48*n^3 - 52*n)*x + (96*n^2 - 76*n + 20)*x^2 + (48*n - 16)*x^3)*v[i - 1] - ((8*n^2 + 4*n)*x + (32*n^2 + 32*n - 2)*x^2 + 64*n*x^3 + 32*x^4)*deriv(v[i - 1], x))/(4*n^3 - 4*n^2 + n + (8*n^2 - 9*n)*x + 4*n*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. [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