Real-rooted atlas

A290315 → D

nonnegative coefficients

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

Recurrence

\(\left(-11 + 4\,n\right)\,P_{n}(x) = \left(21 - 364\,x + 4004\,x^{2} - 12\,n + 320\,n x - 4976\,n x^{2} - 64\,n^{2} x + 1984\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(14\,x - 336\,x^{2} + 616\,x^{3} - 8\,n x + 304\,n x^{2} - 576\,n x^{3} - 64\,n^{2} x^{2} + 128\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-40 + 396\,x + 16\,n - 320\,n x + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-22\,x + 44\,x^{2} + 8\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
3162440
466568360994
5224574417856672030544
6706421603490085842561512001127330
7216026445645932161921862420651520399168048721656
8653015210004881324040569056010439582407953465601210809602416417090

All 15 rows

Technical details
Node id
n-cc6f0733a6e78a5c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2fd84d8d5849b0ab

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 -2)
  • 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:     0;
   2:     2;
   3:    16,       24;
   4:    66,      568,        360;
   5:   224,     5744,      17856,        6720;
   6:   706,    42160,     349008,      584256,       151200;
   7:  2160,   264456,    4593216,    19218624,     20651520,       3991680;
   8:  6530,  1521000,   48813240,   405690560,   1043958240,     795346560,     121080960;
   9: 19648,  8310240,  455767680,  6645093760,  32782295040,   57944001024,   33364961280,    4151347200;
  10: 59010, 43987552, 3913364640, 92621116800, 775923605440, 2562854699520, 3339730469376, 1519565506560, 158789030400;
%F Formula
The row polynomials satisfy (4*n-11)*P(n,x) = -(12*n-21+(64*n^2-320*n+364)*x+(256*n^3-1984*n^2+4976*n-4004)*x^2)*P(n-2,x) - ((8*n-14)*x+(64*n^2-304*n+336)*x^2-(128*n^2-576*n+616)*x^3)*d/dx P(n-2,x) + (16*n-40+(64*n^2-320*n+396)*x)*P(n-1,x) + ((8*n-22)*x-(16*n-44)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 21 + (64*n^2 - 320*n + 364)*x + (256*n^3 - 1984*n^2 + 4976*n - 4004)*x^2)*P[n - 2, x] - ((8*n - 14)*x + (64*n^2 - 304*n + 336)*x^2 - (128*n^2 - 576*n + 616)*x^3)*D[P[n - 2, x], x] + (16*n - 40 + (64*n^2 - 320*n + 396)*x)*P[n - 1, x] + ((8*n - 22)*x - (16*n - 44)*x^2)*D[P[n - 1, x], x])/(4*n - 11)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(12*n - 21 + (64*n^2 - 320*n + 364)*x + (256*n^3 - 1984*n^2 + 4976*n - 4004)*x^2)*v[i - 2] - ((8*n - 14)*x + (64*n^2 - 304*n + 336)*x^2 - (128*n^2 - 576*n + 616)*x^3)*deriv(v[i - 2], x) + (16*n - 40 + (64*n^2 - 320*n + 396)*x)*v[i - 1] + ((8*n - 22)*x - (16*n - 44)*x^2)*deriv(v[i - 1], x))/(4*n - 11)); 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

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.